2010

• Paper 1, Section I, D

  #Analysis I

  2010

  Let $\sum_{n \geqslant 0} a_{n} z^{n}$ be a complex power series. State carefully what it means for the power series to have radius of convergence $R$, with $R \in[0, \infty]$.

  Suppose the power series has radius of convergence $R$, with $0<R<\infty$. Show that the sequence $\left|a_{n} z^{n}\right|$ is unbounded if $|z|>R$.

  Find the radius of convergence of $\sum_{n \geqslant 1} z^{n} / n^{3}$.

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• Paper 1, Section I, E

  #Analysis I

  2010

  Find the limit of each of the following sequences; justify your answers.

  (i)

  $$\frac{1+2+\ldots+n}{n^{2}}$$

  (ii)

  $$\sqrt[n]{n}$$

  (iii)

  $$\left(a^{n}+b^{n}\right)^{1 / n} \quad \text { with } \quad 0<a \leqslant b$$

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• Paper 1, Section II, E

  #Analysis I

  2010

  Determine whether the following series converge or diverge. Any tests that you use should be carefully stated.

  (a)

  $$\sum_{n \geqslant 1} \frac{n !}{n^{n}}$$

  (b)

  $$\sum_{n \geqslant 1} \frac{1}{n+(\log n)^{2}}$$

  (c)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{1+\sqrt{n}}$$

  (d)

  $$\sum_{n \geqslant 1} \frac{(-1)^{n}}{n\left(2+(-1)^{n}\right)}$$

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• Paper 1, Section II, F

  #Analysis I

  2010

  (a) State and prove Taylor's theorem with the remainder in Lagrange's form.

  (b) Suppose that $e: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $e(0)=1$ and $e^{\prime}(x)=e(x)$ for all $x \in \mathbb{R}$. Use the result of (a) to prove that

  $$e(x)=\sum_{n \geqslant 0} \frac{x^{n}}{n !} \quad \text { for all } \quad x \in \mathbb{R}$$

  [No property of the exponential function may be assumed.]

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• Paper 1, Section II, D

  #Analysis I

  2010

  Define what it means for a bounded function $f:[a, \infty) \rightarrow \mathbb{R}$ to be Riemann integrable.

  Show that a monotonic function $f:[a, b] \rightarrow \mathbb{R}$ is Riemann integrable, where $-\infty<a<b<\infty$.

  Prove that if $f:[1, \infty) \rightarrow \mathbb{R}$ is a decreasing function with $f(x) \rightarrow 0$ as $x \rightarrow \infty$, then $\sum_{n \geqslant 1} f(n)$ and $\int_{1}^{\infty} f(x) d x$ either both diverge or both converge.

  Hence determine, for $\alpha \in \mathbb{R}$, when $\sum_{n \geqslant 1} n^{\alpha}$ converges.

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• Paper 1, Section II, F

  #Analysis I

  2010

  (a) Let $n \geqslant 1$ and $f$ be a function $\mathbb{R} \rightarrow \mathbb{R}$. Define carefully what it means for $f$ to be $n$ times differentiable at a point $x_{0} \in \mathbb{R}$.

  $$\text { Set } \operatorname{sign}(x)= \begin{cases}x /|x|, & x \neq 0 \\ 0, & x=0 .\end{cases}$$

  Consider the function $f(x)$ on the real line, with $f(0)=0$ and

  $$f(x)=x^{2} \operatorname{sign}(x)\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 .$$

  (b) Is $f(x)$ differentiable at $x=0$ ?

  (c) Show that $f(x)$ has points of non-differentiability in any neighbourhood of $x=0$.

  (d) Prove that, in any finite interval $I$, the derivative $f^{\prime}(x)$, at the points $x \in I$ where it exists, is bounded: $\left|f^{\prime}(x)\right| \leqslant C$ where $C$ depends on $I$.

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• Paper 2, Section I, A

  #Differential Equations

  2010

  Find the general solutions to the following difference equations for $y_{n}, n \in \mathbb{N}$.

  $$\begin{aligned} \text { (i) } & y_{n+3}-3 y_{n+1}+2 y_{n}=0, \\ \text { (ii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=2^{n} \\ \text { (iii) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n} \\ \text { (iv) } & y_{n+3}-3 y_{n+1}+2 y_{n}=(-2)^{n}+2^{n} . \end{aligned}$$

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• Paper 2, Section I, A

  #Differential Equations

  2010

  Let $f(x, y)=g(u, v)$ where the variables $\{x, y\}$ and $\{u, v\}$ are related by a smooth, invertible transformation. State the chain rule expressing the derivatives $\frac{\partial g}{\partial u}$ and $\frac{\partial g}{\partial v}$ in terms of $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ and use this to deduce that

  $$\frac{\partial^{2} g}{\partial u \partial v}=\frac{\partial x}{\partial u} \frac{\partial x}{\partial v} \frac{\partial^{2} f}{\partial x^{2}}+\left(\frac{\partial x}{\partial u} \frac{\partial y}{\partial v}+\frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\right) \frac{\partial^{2} f}{\partial x \partial y}+\frac{\partial y}{\partial u} \frac{\partial y}{\partial v} \frac{\partial^{2} f}{\partial y^{2}}+H \frac{\partial f}{\partial x}+K \frac{\partial f}{\partial y}$$

  where $H$ and $K$ are second-order partial derivatives, to be determined.

  Using the transformation $x=u v$ and $y=u / v$ in the above identity, or otherwise, find the general solution of

  $$x \frac{\partial^{2} f}{\partial x^{2}}-\frac{y^{2}}{x} \frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial f}{\partial x}-\frac{y}{x} \frac{\partial f}{\partial y}=0$$

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• Paper 2, Section II, A

  #Differential Equations

  2010

  (a) Find the general solution of the system of differential equations

  $$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$

  (b) Depending on the parameter $\lambda \in \mathbb{R}$, find the general solution of the system of differential equations

  $$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)+2\left(\begin{array}{r} -\lambda \\ 1 \\ \lambda \end{array}\right) e^{2 t},$$

  and explain why $(2)$ has a particular solution of the form $\mathbf{c} e^{2 t}$ with constant vector $\mathbf{c} \in \mathbb{R}^{3}$ for $\lambda=1$ but not for $\lambda \neq 1$.

  [Hint: decompose $\left(\begin{array}{c}-\lambda \\ 1 \\ \lambda\end{array}\right)$ in terms of the eigenbasis of the matrix in (1).]

  (c) For $\lambda=-1$, find the solution of (2) which goes through the point $(0,1,0)$ at $t=0$.

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• Paper 2, Section II, A

  #Differential Equations

  2010

  (a) Consider the differential equation

  $$a_{n} \frac{d^{n} y}{d x^{n}}+a_{n-1} \frac{d^{n-1} y}{d x^{n-1}}+\ldots+a_{2} \frac{d^{2} y}{d x^{2}}+a_{1} \frac{d y}{d x}+a_{0} y=0$$

  with $n \in \mathbb{N}$ and $a_{0}, \ldots, a_{n} \in \mathbb{R}$. Show that $y(x)=e^{\lambda x}$ is a solution if and only if $p(\lambda)=0$ where

  $$p(\lambda)=a_{n} \lambda^{n}+a_{n-1} \lambda^{n-1}+\ldots+a_{2} \lambda^{2}+a_{1} \lambda+a_{0}$$

  Show further that $y(x)=x e^{\mu x}$ is also a solution of $(1)$ if $\mu$ is a root of the polynomial $p(\lambda)$ of multiplicity at least 2 .

  (b) By considering $v(t)=\frac{d^{2} u}{d t^{2}}$, or otherwise, find the general real solution for $u(t)$ satisfying

  $$\frac{d^{4} u}{d t^{4}}+2 \frac{d^{2} u}{d t^{2}}=4 t^{2}$$

  By using a substitution of the form $u(t)=y\left(t^{2}\right)$ in $(2)$, or otherwise, find the general real solution for $y(x)$, with $x$ positive, where

  $$4 x^{2} \frac{d^{4} y}{d x^{4}}+12 x \frac{d^{3} y}{d x^{3}}+(3+2 x) \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}=x$$

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• Paper 2, Section II, $6 \mathrm{~A}$

  #Differential Equations

  2010

  (a) By using a power series of the form

  $$y(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$

  or otherwise, find the general solution of the differential equation

  $$x y^{\prime \prime}-(1-x) y^{\prime}-y=0 .$$

  (b) Define the Wronskian $W(x)$ for a second order linear differential equation

  $$y^{\prime \prime}+p(x) y^{\prime}+q(x) y=0$$

  and show that $W^{\prime}+p(x) W=0$. Given a non-trivial solution $y_{1}(x)$ of $(2)$ show that $W(x)$ can be used to find a second solution $y_{2}(x)$ of $(2)$ and give an expression for $y_{2}(x)$ in the form of an integral.

  (c) Consider the equation (2) with

  $$p(x)=-\frac{P(x)}{x} \quad \text { and } \quad q(x)=-\frac{Q(x)}{x}$$

  where $P$ and $Q$ have Taylor expansions

  $$P(x)=P_{0}+P_{1} x+\ldots, \quad Q(x)=Q_{0}+Q_{1} x+\ldots$$

  with $P_{0}$ a positive integer. Find the roots of the indicial equation for (2) with these assumptions. If $y_{1}(x)=1+\beta x+\ldots$ is a solution, use the method of part (b) to find the first two terms in a power series expansion of a linearly independent solution $y_{2}(x)$, expressing the coefficients in terms of $P_{0}, P_{1}$ and $\beta$.

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• Paper 2, Section II, A

  #Differential Equations

  2010

  (a) State how the nature of a critical (or stationary) point of a function $f(\mathbf{x})$ with $\mathbf{x} \in \mathbb{R}^{n}$ can be determined by consideration of the eigenvalues of the Hessian matrix $H$ of $f(\mathbf{x})$, assuming $H$ is non-singular.

  (b) Let $f(x, y)=x y(1-x-y)$. Find all the critical points of the function $f(x, y)$ and determine their nature. Determine the zero contour of $f(x, y)$ and sketch a contour plot showing the behaviour of the contours in the neighbourhood of the critical points.

  (c) Now let $g(x, y)=x^{3} y^{2}(1-x-y)$. Show that $(0,1)$ is a critical point of $g(x, y)$ for which the Hessian matrix of $g$ is singular. Find an approximation for $g(x, y)$ to lowest non-trivial order in the neighbourhood of the point $(0,1)$. Does $g$ have a maximum or a minimum at $(0,1)$ ? Justify your answer.

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• Paper 4, Section I, B

  #Dynamics and Relativity

  2010

  A particle of mass $m$ and charge $q$ moves with trajectory $\mathbf{r}(t)$ in a constant magnetic field $\mathbf{B}=B \hat{\mathbf{z}}$. Write down the Lorentz force on the particle and use Newton's Second Law to deduce that

  $$\dot{\mathbf{r}}-\omega \mathbf{r} \times \hat{\mathbf{z}}=\mathbf{c}$$

  where $\mathbf{c}$ is a constant vector and $\omega$ is to be determined. Find $\mathbf{c}$ and hence $\mathbf{r}(t)$ for the initial conditions

  $$\mathbf{r}(0)=a \hat{\mathbf{x}} \quad \text { and } \quad \dot{\mathbf{r}}(0)=u \hat{\mathbf{y}}+v \hat{\mathbf{z}}$$

  where $a, u$ and $v$ are constants. Sketch the particle's trajectory in the case $a \omega+u=0$.

  [Unit vectors $\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}$ correspond to a set of Cartesian coordinates. ]

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• Paper 4, Section $I$, B

  #Dynamics and Relativity

  2010

  Let $S$ be an inertial frame with coordinates $(t, x)$ in two-dimensional spacetime. Write down the Lorentz transformation giving the coordinates $\left(t^{\prime}, x^{\prime}\right)$ in a second inertial frame $S^{\prime}$ moving with velocity $v$ relative to $S$. If a particle has constant velocity $u$ in $S$, find its velocity $u^{\prime}$ in $S^{\prime}$. Given that $|u|<c$ and $|v|<c$, show that $\left|u^{\prime}\right|<c$.

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• Paper 4 , Section II, B

  #Dynamics and Relativity

  2010

  A sphere of uniform density has mass $m$ and radius $a$. Find its moment of inertia about an axis through its centre.

  A marble of uniform density is released from rest on a plane inclined at an angle $\alpha$ to the horizontal. Let the time taken for the marble to travel a distance $\ell$ down the plane be: (i) $t_{1}$ if the plane is perfectly smooth; or (ii) $t_{2}$ if the plane is rough and the marble rolls without slipping.

  Explain, with a clear discussion of the forces acting on the marble, whether or not its energy is conserved in each of the cases (i) and (ii). Show that $t_{1} / t_{2}=\sqrt{5 / 7}$.

  Suppose that the original marble is replaced by a new one with the same mass and radius but with a hollow centre, so that its moment of inertia is $\lambda m a^{2}$ for some constant $\lambda$. What is the new value for $t_{1} / t_{2}$ ?

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2010

  A particle of unit mass moves in a plane with polar coordinates $(r, \theta)$ and components of acceleration $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$. The particle experiences a force corresponding to a potential $-Q / r$. Show that

  $$E=\frac{1}{2} \dot{r}^{2}+U(r) \quad \text { and } \quad h=r^{2} \dot{\theta}$$

  are constants of the motion, where

  $$U(r)=\frac{h^{2}}{2 r^{2}}-\frac{Q}{r}$$

  Sketch the graph of $U(r)$ in the cases $Q>0$ and $Q<0$.

  (a) Assuming $Q>0$ and $h>0$, for what range of values of $E$ do bounded orbits exist? Find the minimum and maximum distances from the origin, $r_{\min }$ and $r_{\max }$, on such an orbit and show that

  $$r_{\min }+r_{\max }=\frac{Q}{|E|} .$$

  Prove that the minimum and maximum values of the particle's speed, $v_{\min }$ and $v_{\max }$, obey

  $$v_{\min }+v_{\max }=\frac{2 Q}{h}$$

  (b) Now consider trajectories with $E>0$ and $Q$ of either sign. Find the distance of closest approach, $r_{\min }$, in terms of the impact parameter, $b$, and $v_{\infty}$, the limiting value of the speed as $r \rightarrow \infty$. Deduce that if $b \ll|Q| / v_{\infty}^{2}$ then, to leading order,

  $$r_{\min } \approx \frac{2|Q|}{v_{\infty}^{2}} \text { for } Q<0, \quad r_{\min } \approx \frac{b^{2} v_{\infty}^{2}}{2 Q} \text { for } Q>0$$

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2010

  Consider a set of particles with position vectors $\mathbf{r}_{i}(t)$ and masses $m_{i}$, where $i=1,2, \ldots, N$. Particle $i$ experiences an external force $\mathbf{F}_{i}$ and an internal force $\mathbf{F}_{i j}$ from particle $j$, for each $j \neq i$. Stating clearly any assumptions you need, show that

  $$\frac{d \mathbf{P}}{d t}=\mathbf{F} \quad \text { and } \quad \frac{d \mathbf{L}}{d t}=\mathbf{G}$$

  where $\mathbf{P}$ is the total momentum, $\mathbf{F}$ is the total external force, $\mathbf{L}$ is the total angular momentum about a fixed point $\mathbf{a}$, and $\mathbf{G}$ is the total external torque about $\mathbf{a}$.

  Does the result $\frac{d \mathbf{L}}{d t}=\mathbf{G}$ still hold if the fixed point $\mathbf{a}$ is replaced by the centre of mass of the system? Justify your answer.

  Suppose now that the external force on particle $i$ is $-k \frac{d \mathbf{r}_{i}}{d t}$ and that all the particles have the same mass $m$. Show that

  $$\mathbf{L}(t)=\mathbf{L}(0) e^{-k t / m}$$

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• Paper 4, Section II, B

  #Dynamics and Relativity

  2010

  A particle $A$ of rest mass $m$ is fired at an identical particle $B$ which is stationary in the laboratory. On impact, $A$ and $B$ annihilate and produce two massless photons whose energies are equal. Assuming conservation of four-momentum, show that the angle $\theta$ between the photon trajectories is given by

  $$\cos \theta=\frac{E-3 m c^{2}}{E+m c^{2}}$$

  where $E$ is the relativistic energy of $A$.

  Let $v$ be the speed of the incident particle $A$. For what value of $v / c$ will the photons move in perpendicular directions? If $v$ is very small compared with $c$, show that

  $$\theta \approx \pi-v / c$$

  [All quantities referred to are measured in the laboratory frame.]

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• Paper 3, Section I, D

  #Groups

  2010

  Write down the matrix representing the following transformations of $\mathbb{R}^{3}$ :

  (i) clockwise rotation of $45^{\circ}$ around the $x$ axis,

  (ii) reflection in the plane $x=y$,

  (iii) the result of first doing (i) and then (ii).

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• Paper 3, Section I, D

  #Groups

  2010

  Express the element $(123)(234)$ in $S_{5}$ as a product of disjoint cycles. Show that it is in $A_{5}$. Write down the elements of its conjugacy class in $A_{5}$.

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• Paper 3, Section II, D

  #Groups

  2010

  (i) State the orbit-stabilizer theorem.

  Let $G$ be the group of rotations of the cube, $X$ the set of faces. Identify the stabilizer of a face, and hence compute the order of $G$.

  Describe the orbits of $G$ on the set $X \times X$ of pairs of faces.

  (ii) Define what it means for a subgroup $N$ of $G$ to be normal. Show that $G$ has a normal subgroup of order 4 .

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• Paper 3, Section II, D

  #Groups

  2010

  State Lagrange's theorem. Let $p$ be a prime number. Prove that every group of order $p$ is cyclic. Prove that every abelian group of order $p^{2}$ is isomorphic to either $C_{p} \times C_{p}$ or $C_{p^{2} \text {. }}$

  Show that $D_{12}$, the dihedral group of order 12 , is not isomorphic to the alternating $\operatorname{group} A_{4}$.

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• Paper 3, Section II, D

  #Groups

  2010

  Let $G$ be a group, $X$ a set on which $G$ acts transitively, $B$ the stabilizer of a point $x \in X$.

  Show that if $g \in G$ stabilizes the point $y \in X$, then there exists an $h \in G$ with $h g h^{-1} \in B$.

  Let $G=S L_{2}(\mathbb{C})$, acting on $\mathbb{C} \cup\{\infty\}$ by Möbius transformations. Compute $B=G_{\infty}$, the stabilizer of $\infty$. Given

  $$g=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \in G$$

  compute the set of fixed points $\{x \in \mathbb{C} \cup\{\infty\} \mid g x=x\} .$

  Show that every element of $G$ is conjugate to an element of $B$.

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• Paper 3, Section II, D

  #Groups

  2010

  Let $G$ be a finite group, $X$ the set of proper subgroups of $G$. Show that conjugation defines an action of $G$ on $X$.

  Let $B$ be a proper subgroup of $G$. Show that the orbit of $G$ on $X$ containing $B$ has size at most the index $|G: B|$. Show that there exists a $g \in G$ which is not conjugate to an element of $B$.

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• Paper 4, Section I, E

  #Numbers and Sets

  2010

  (a) Find the smallest residue $x$ which equals $28 ! 13^{28}(\bmod 31)$.

  [You may use any standard theorems provided you state them correctly.]

  (b) Find all integers $x$ which satisfy the system of congruences

  $$\begin{aligned} x & \equiv 1(\bmod 2) \\ 2 x & \equiv 1(\bmod 3) \\ 2 x & \equiv 4(\bmod 10) \\ x & \equiv 10(\bmod 67) \end{aligned}$$

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• Paper 4, Section I, $2 E$

  #Numbers and Sets

  2010

  (a) Let $r$ be a real root of the polynomial $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}$, with integer coefficients $a_{i}$ and leading coefficient 1 . Show that if $r$ is rational, then $r$ is an integer.

  (b) Write down a series for $e$. By considering $q ! e$ for every natural number $q$, show that $e$ is irrational.

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• Paper 4, Section II, E

  #Numbers and Sets

  2010

  The Fibonacci numbers $F_{n}$ are defined for all natural numbers $n$ by the rules

  $$F_{1}=1, \quad F_{2}=1, \quad F_{n}=F_{n-1}+F_{n-2} \quad \text { for } n \geqslant 3$$

  Prove by induction on $k$ that, for any $n$,

  $$F_{n+k}=F_{k} F_{n+1}+F_{k-1} F_{n} \text { for all } k \geqslant 2 \text {. }$$

  Deduce that

  $$F_{2 n}=F_{n}\left(F_{n+1}+F_{n-1}\right) \quad \text { for all } n \geqslant 2$$

  Put $L_{1}=1$ and $L_{n}=F_{n+1}+F_{n-1}$ for $n>1$. Show that these (Lucas) numbers $L_{n}$ satisfy

  $$L_{1}=1, \quad L_{2}=3, \quad L_{n}=L_{n-1}+L_{n-2} \quad \text { for } n \geqslant 3$$

  Show also that, for all $n$, the greatest common divisor $\left(F_{n}, F_{n+1}\right)$ is 1 , and that the greatest common divisor $\left(F_{n}, L_{n}\right)$ is at most 2 .

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• Paper 4, Section II, $6 \mathrm{E}$

  #Numbers and Sets

  2010

  State and prove Fermat's Little Theorem.

  Let $p$ be an odd prime. If $p \neq 5$, show that $p$ divides $10^{n}-1$ for infinitely many natural numbers $n$.

  Hence show that $p$ divides infinitely many of the integers

  $$5,55, \quad 555, \quad 5555, \quad \ldots .$$

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• Paper 4, Section II, $7 \mathrm{E}$

  #Numbers and Sets

  2010

  (a) Let $A, B$ be finite non-empty sets, with $|A|=a,|B|=b$. Show that there are $b^{a}$ mappings from $A$ to $B$. How many of these are injective ?

  (b) State the Inclusion-Exclusion principle.

  (c) Prove that the number of surjective mappings from a set of size $n$ onto a set of size $k$ is

  $$\sum_{i=0}^{k}(-1)^{i}\left(\begin{array}{c} k \\ i \end{array}\right)(k-i)^{n} \quad \text { for } n \geqslant k \geqslant 1$$

  Deduce that

  $$n !=\sum_{i=0}^{n}(-1)^{i}\left(\begin{array}{c} n \\ i \end{array}\right)(n-i)^{n}$$

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• Paper 4 , Section II, E

  #Numbers and Sets

  2010

  What does it mean for a set to be countable ?

  Show that $\mathbb{Q}$ is countable, but $\mathbb{R}$ is not. Show also that the union of two countable sets is countable.

  A subset $A$ of $\mathbb{R}$ has the property that, given $\epsilon>0$ and $x \in \mathbb{R}$, there exist reals $a, b$ with $a \in A$ and $b \notin A$ with $|x-a|<\epsilon$ and $|x-b|<\epsilon$. Can $A$ be countable ? Can $A$ be uncountable ? Justify your answers.

  A subset $B$ of $\mathbb{R}$ has the property that given $b \in B$ there exists $\epsilon>0$ such that if $0<|b-x|<\epsilon$ for some $x \in \mathbb{R}$, then $x \notin B$. Is $B$ countable ? Justify your answer.

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• Paper 2, Section I, F

  #Probability

  2010

  Jensen's inequality states that for a convex function $f$ and a random variable $X$ with a finite mean, $\mathbb{E} f(X) \geqslant f(\mathbb{E} X)$.

  (a) Suppose that $f(x)=x^{m}$ where $m$ is a positive integer, and $X$ is a random variable taking values $x_{1}, \ldots, x_{N} \geqslant 0$ with equal probabilities, and where the sum $x_{1}+\ldots+x_{N}=1$. Deduce from Jensen's inequality that

  $$\sum_{i=1}^{N} f\left(x_{i}\right) \geqslant N f\left(\frac{1}{N}\right)$$

  (b) $N$ horses take part in $m$ races. The results of different races are independent. The probability for horse $i$ to win any given race is $p_{i} \geqslant 0$, with $p_{1}+\ldots+p_{N}=1$.

  Let $Q$ be the probability that a single horse wins all $m$ races. Express $Q$ as a polynomial of degree $m$ in the variables $p_{1}, \ldots, p_{N}$.

  By using (1) or otherwise, prove that $Q \geqslant N^{1-m}$.

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• Paper 2, Section I, F

  #Probability

  2010

  Let $X$ and $Y$ be two non-constant random variables with finite variances. The correlation coefficient $\rho(X, Y)$ is defined by

  $$\rho(X, Y)=\frac{\mathbb{E}[(X-\mathbb{E} X)(Y-\mathbb{E} Y)]}{(\operatorname{Var} X)^{1 / 2}(\operatorname{Var} Y)^{1 / 2}}$$

  (a) Using the Cauchy-Schwarz inequality or otherwise, prove that

  $$-1 \leqslant \rho(X, Y) \leqslant 1$$

  (b) What can be said about the relationship between $X$ and $Y$ when either (i) $\rho(X, Y)=0$ or (ii) $|\rho(X, Y)|=1$. [Proofs are not required.]

  (c) Take $0 \leqslant r \leqslant 1$ and let $X, X^{\prime}$ be independent random variables taking values $\pm 1$ with probabilities $1 / 2$. Set

  $$Y= \begin{cases}X, & \text { with probability } r \\ X^{\prime}, & \text { with probability } 1-r\end{cases}$$

  Find $\rho(X, Y)$.

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• Paper 2, Section II, F

  #Probability

  2010

  (a) What does it mean to say that a random variable $X$ with values $n=1,2, \ldots$ has a geometric distribution with a parameter $p$ where $p \in(0,1)$ ?

  An expedition is sent to the Himalayas with the objective of catching a pair of wild yaks for breeding. Assume yaks are loners and roam about the Himalayas at random. The probability $p \in(0,1)$ that a given trapped yak is male is independent of prior outcomes. Let $N$ be the number of yaks that must be caught until a breeding pair is obtained. (b) Find the expected value of $N$. (c) Find the variance of $N$.

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• Paper 2, Section II, F

  #Probability

  2010

  The yearly levels of water in the river Camse are independent random variables $X_{1}, X_{2}, \ldots$, with a given continuous distribution function $F(x)=\mathbb{P}\left(X_{i} \leqslant x\right), x \geqslant 0$ and $F(0)=0$. The levels have been observed in years $1, \ldots, n$ and their values $X_{1}, \ldots, X_{n}$ recorded. The local council has decided to construct a dam of height

  $$Y_{n}=\max \left[X_{1}, \ldots, X_{n}\right]$$

  Let $\tau$ be the subsequent time that elapses before the dam overflows:

  $$\tau=\min \left[t \geqslant 1: X_{n+t}>Y_{n}\right]$$

  (a) Find the distribution function $\mathbb{P}\left(Y_{n} \leqslant z\right), z>0$, and show that the mean value $\mathbb{E} Y_{n}=\int_{0}^{\infty}\left[1-F(z)^{n}\right] \mathrm{d} z .$

  (b) Express the conditional probability $\mathbb{P}\left(\tau=k \mid Y_{n}=z\right)$, where $k=1,2, \ldots$ and $z>0$, in terms of $F$.

  (c) Show that the unconditional probability

  $$\mathbb{P}(\tau=k)=\frac{n}{(k+n-1)(k+n)}, \quad k=1,2, \ldots$$

  (d) Determine the mean value $\mathbb{E} \tau$.

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• Paper 2, Section II, F

  #Probability

  2010

  In a branching process every individual has probability $p_{k}$ of producing exactly $k$ offspring, $k=0,1, \ldots$, and the individuals of each generation produce offspring independently of each other and of individuals in preceding generations. Let $X_{n}$ represent the size of the $n$th generation. Assume that $X_{0}=1$ and $p_{0}>0$ and let $F_{n}(s)$ be the generating function of $X_{n}$. Thus

  $$F_{1}(s)=\mathbb{E} s^{X_{1}}=\sum_{k=0}^{\infty} p_{k} s^{k},|s| \leqslant 1$$

  (a) Prove that

  $$F_{n+1}(s)=F_{n}\left(F_{1}(s)\right)$$

  (b) State a result in terms of $F_{1}(s)$ about the probability of eventual extinction. [No proofs are required.]

  (c) Suppose the probability that an individual leaves $k$ descendants in the next generation is $p_{k}=1 / 2^{k+1}$, for $k \geqslant 0$. Show from the result you state in (b) that extinction is certain. Prove further that in this case

  $$F_{n}(s)=\frac{n-(n-1) s}{(n+1)-n s}, \quad n \geqslant 1$$

  and deduce the probability that the $n$th generation is empty.

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• Paper 2, Section II, F

  #Probability

  2010

  Let $X_{1}, X_{2}$ be bivariate normal random variables, with the joint probability density function

  $$f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right)=\frac{1}{2 \pi \sigma_{1} \sigma_{2} \sqrt{1-\rho^{2}}} \exp \left[-\frac{\varphi\left(x_{1}, x_{2}\right)}{2\left(1-\rho^{2}\right)}\right]$$

  where

  $$\varphi\left(x_{1}, x_{2}\right)=\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)^{2}-2 \rho\left(\frac{x_{1}-\mu_{1}}{\sigma_{1}}\right)\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)+\left(\frac{x_{2}-\mu_{2}}{\sigma_{2}}\right)^{2}$$

  and $x_{1}, x_{2} \in \mathbb{R}$.

  (a) Deduce that the marginal probability density function

  $$f_{X_{1}}\left(x_{1}\right)=\frac{1}{\sqrt{2 \pi} \sigma_{1}} \exp \left[-\frac{\left(x_{1}-\mu_{1}\right)^{2}}{2 \sigma_{1}^{2}}\right]$$

  (b) Write down the moment-generating function of $X_{2}$ in terms of $\mu_{2}$ and $\sigma_{2} \cdot[N o$ proofs are required.]

  (c) By considering the ratio $f_{X_{1}, X_{2}}\left(x_{1}, x_{2}\right) / f_{X_{2}}\left(x_{2}\right)$ prove that, conditional on $X_{2}=x_{2}$, the distribution of $X_{1}$ is normal, with mean and variance $\mu_{1}+\rho \sigma_{1}\left(x_{2}-\mu_{2}\right) / \sigma_{2}$ and $\sigma_{1}^{2}\left(1-\rho^{2}\right)$, respectively.

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• Paper 3, Section I, C

  #Vector Calculus

  2010

  Consider the vector field

  $$\mathbf{F}=\left(-y /\left(x^{2}+y^{2}\right), x /\left(x^{2}+y^{2}\right), 0\right)$$

  defined on all of $\mathbb{R}^{3}$ except the $z$ axis. Compute $\boldsymbol{\nabla} \times \mathbf{F}$ on the region where it is defined.

  Let $\gamma_{1}$ be the closed curve defined by the circle in the $x y$-plane with centre $(2,2,0)$ and radius 1 , and $\gamma_{2}$ be the closed curve defined by the circle in the $x y$-plane with centre $(0,0,0)$ and radius 1 .

  By using your earlier result, or otherwise, evaluate the line integral $\oint_{\gamma_{1}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$.

  By explicit computation, evaluate the line integral $\oint_{\gamma_{2}} \mathbf{F} \cdot \mathrm{d} \mathbf{x}$. Is your result consistent with Stokes' theorem? Explain your answer briefly.

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• Paper 3, Section I, C

  #Vector Calculus

  2010

  A curve in two dimensions is defined by the parameterised Cartesian coordinates

  $$x(u)=a e^{b u} \cos u, \quad y(u)=a e^{b u} \sin u$$

  where the constants $a, b>0$. Sketch the curve segment corresponding to the range $0 \leqslant u \leqslant 3 \pi$. What is the length of the curve segment between the points $(x(0), y(0))$ and $(x(U), y(U))$, as a function of $U$ ?

  A geometrically sensitive ant walks along the curve with varying speed $\kappa(u)^{-1}$, where $\kappa(u)$ is the curvature at the point corresponding to parameter $u$. Find the time taken by the ant to walk from $(x(2 n \pi), y(2 n \pi))$ to $(x(2(n+1) \pi), y(2(n+1) \pi))$, where $n$ is a positive integer, and hence verify that this time is independent of $n$.

  [You may quote without proof the formula $\kappa(u)=\frac{\left|x^{\prime}(u) y^{\prime \prime}(u)-y^{\prime}(u) x^{\prime \prime}(u)\right|}{\left(\left(x^{\prime}(u)\right)^{2}+\left(y^{\prime}(u)\right)^{2}\right)^{3 / 2}} .$ ]

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• Paper 3, Section II, C

  #Vector Calculus

  2010

  (a) Define a rank two tensor and show that if two rank two tensors $A_{i j}$ and $B_{i j}$ are the same in one Cartesian coordinate system, then they are the same in all Cartesian coordinate systems.

  The quantity $C_{i j}$ has the property that, for every rank two tensor $A_{i j}$, the quantity $C_{i j} A_{i j}$ is a scalar. Is $C_{i j}$ necessarily a rank two tensor? Justify your answer with a proof from first principles, or give a counterexample.

  (b) Show that, if a tensor $T_{i j}$ is invariant under rotations about the $x_{3}$-axis, then it has the form

  $$\left(\begin{array}{ccc} \alpha & \omega & 0 \\ -\omega & \alpha & 0 \\ 0 & 0 & \beta \end{array}\right)$$

  (c) The inertia tensor about the origin of a rigid body occupying volume $V$ and with variable mass density $\rho(\mathbf{x})$ is defined to be

  $$I_{i j}=\int_{V} \rho(\mathbf{x})\left(x_{k} x_{k} \delta_{i j}-x_{i} x_{j}\right) \mathrm{d} V$$

  The rigid body $B$ has uniform density $\rho$ and occupies the cylinder

  $$\left\{\left(x_{1}, x_{2}, x_{3}\right):-2 \leqslant x_{3} \leqslant 2, x_{1}^{2}+x_{2}^{2} \leqslant 1\right\}$$

  Show that the inertia tensor of $B$ about the origin is diagonal in the $\left(x_{1}, x_{2}, x_{3}\right)$ coordinate system, and calculate its diagonal elements.

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• Paper 3, Section II, C

  #Vector Calculus

  2010

  Let $f(x, y)$ be a function of two variables, and $R$ a region in the $x y$-plane. State the rule for evaluating $\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y$ as an integral with respect to new variables $u(x, y)$ and $v(x, y)$.

  Sketch the region $R$ in the $x y$-plane defined by

  $$R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}$$

  Sketch the corresponding region in the $u v$-plane, where

  $$u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}$$

  Express the integral

  $$I=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y$$

  as an integral with respect to $u$ and $v$. Hence, or otherwise, calculate $I$.

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• Paper 3 , Section II, C

  #Vector Calculus

  2010

  State the divergence theorem (also known as Gauss' theorem) relating the surface and volume integrals of appropriate fields.

  The surface $S_{1}$ is defined by the equation $z=3-2 x^{2}-2 y^{2}$ for $1 \leqslant z \leqslant 3$; the surface $S_{2}$ is defined by the equation $x^{2}+y^{2}=1$ for $0 \leqslant z \leqslant 1$; the surface $S_{3}$ is defined by the equation $z=0$ for $x, y$ satisfying $x^{2}+y^{2} \leqslant 1$. The surface $S$ is defined to be the union of the surfaces $S_{1}, S_{2}$ and $S_{3}$. Sketch the surfaces $S_{1}, S_{2}, S_{3}$ and (hence) $S$.

  The vector field $\mathbf{F}$ is defined by

  $$\mathbf{F}(x, y, z)=\left(x y+x^{6},-\frac{1}{2} y^{2}+y^{8}, z\right)$$

  Evaluate the integral

  $$\oint_{S} \mathbf{F} \cdot \mathrm{d} \mathbf{S}$$

  where the surface element $\mathrm{d} \mathbf{S}$ points in the direction of the outward normal to $S$.

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• Paper 3, Section II, C

  #Vector Calculus

  2010

  Given a spherically symmetric mass distribution with density $\rho$, explain how to obtain the gravitational field $\mathbf{g}=-\nabla \phi$, where the potential $\phi$ satisfies Poisson's equation

  $$\nabla^{2} \phi=4 \pi G \rho$$

  The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells $S_{n}$ labelled by integers $n \geqslant 1$. The shell $S_{n}$ has uniform density $2^{n-1} \rho_{0}$, where $\rho_{0}$ is a constant, and occupies the volume between radius $2^{-n+1}$ and $2^{-n}$.

  Obtain a closed form expression for the mass of Geometria.

  Obtain a closed form expression for the gravitational field $\mathbf{g}$ due to Geometria at a distance $r=2^{-N}$ from its centre of mass, for each positive integer $N \geqslant 1$. What is the potential $\phi(r)$ due to Geometria for $r>1$ ?

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• Paper 1, Section I, $1 A$

  #Vectors and Matrices

  2010

  Let $A$ be the matrix representing a linear map $\Phi: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ with respect to the bases $\left\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\}$ of $\mathbb{R}^{n}$ and $\left\{\mathbf{c}_{1}, \ldots, \mathbf{c}_{m}\right\}$ of $\mathbb{R}^{m}$, so that $\Phi\left(\mathbf{b}_{i}\right)=A_{j i} \mathbf{c}_{j}$. Let $\left\{\mathbf{b}_{1}^{\prime}, \ldots, \mathbf{b}_{n}^{\prime}\right\}$ be another basis of $\mathbb{R}^{n}$ and let $\left\{\mathbf{c}_{1}^{\prime}, \ldots, \mathbf{c}_{m}^{\prime}\right\}$ be another basis of $\mathbb{R}^{m}$. Show that the matrix $A^{\prime}$ representing $\Phi$ with respect to these new bases satisfies $A^{\prime}=C^{-1} A B$ with matrices $B$ and $C$ which should be defined.

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• Paper 1, Section I, C

  #Vectors and Matrices

  2010

  (a) The complex numbers $z_{1}$ and $z_{2}$ satisfy the equations

  $$z_{1}^{3}=1, \quad z_{2}^{9}=512 .$$

  What are the possible values of $\left|z_{1}-z_{2}\right|$ ? Justify your answer.

  (b) Show that $\left|z_{1}+z_{2}\right| \leqslant\left|z_{1}\right|+\left|z_{2}\right|$ for all complex numbers $z_{1}$ and $z_{2}$. Does the inequality $\left|z_{1}+z_{2}\right|+\left|z_{1}-z_{2}\right| \leqslant 2 \max \left(\left|z_{1}\right|,\left|z_{2}\right|\right)$ hold for all complex numbers $z_{1}$ and $z_{2}$ ? Justify your answer with a proof or a counterexample.

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• Paper 1, Section II, A

  #Vectors and Matrices

  2010

  Let $A$ and $B$ be real $n \times n$ matrices.

  (i) Define the trace of $A, \operatorname{tr}(A)$, and show that $\operatorname{tr}\left(A^{T} B\right)=\operatorname{tr}\left(B^{T} A\right)$.

  (ii) Show that $\operatorname{tr}\left(A^{T} A\right) \geqslant 0$, with $\operatorname{tr}\left(A^{T} A\right)=0$ if and only if $A$ is the zero matrix. Hence show that

  $$\left(\operatorname{tr}\left(A^{T} B\right)\right)^{2} \leqslant \operatorname{tr}\left(A^{T} A\right) \operatorname{tr}\left(B^{T} B\right)$$

  Under what condition on $A$ and $B$ is equality achieved?

  (iii) Find a basis for the subspace of $2 \times 2$ matrices $X$ such that

  $$\begin{gathered} \operatorname{tr}\left(A^{T} X\right)=\operatorname{tr}\left(B^{T} X\right)=\operatorname{tr}\left(C^{T} X\right)=0 \\ \text { where } \quad A=\left(\begin{array}{ll} 1 & 1 \\ 2 & 0 \end{array}\right), \quad B=\left(\begin{array}{rr} 1 & 1 \\ 0 & -2 \end{array}\right), \quad C=\left(\begin{array}{ll} 0 & 0 \\ 1 & 1 \end{array}\right) \end{gathered}$$

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• Paper 1, Section II, $6 \mathbf{C}$

  #Vectors and Matrices

  2010

  Let $\mathbf{a}_{1}, \mathbf{a}_{2}$ and $\mathbf{a}_{3}$ be vectors in $\mathbb{R}^{3}$. Give a definition of the dot product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2}$, the cross product, $\mathbf{a}_{1} \times \mathbf{a}_{2}$, and the triple product, $\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}$. Explain what it means to say that the three vectors are linearly independent.

  Let $\mathbf{b}_{1}, \mathbf{b}_{2}$ and $\mathbf{b}_{3}$ be vectors in $\mathbb{R}^{3}$. Let $S$ be a $3 \times 3$ matrix with entries $S_{i j}=\mathbf{a}_{i} \cdot \mathbf{b}_{j}$. Show that

  $$\left(\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3}\right)\left(\mathbf{b}_{1} \cdot \mathbf{b}_{2} \times \mathbf{b}_{3}\right)=\operatorname{det}(S)$$

  Hence show that $S$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{a}_{1}, \mathbf{a}_{2}\right.$, $\left.\mathbf{a}_{3}\right\}$ and $\left\{\mathbf{b}_{1}, \mathbf{b}_{2}, \mathbf{b}_{3}\right\}$ are both linearly independent.

  Now let $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ be sets of vectors in $\mathbb{R}^{n}$, and let $T$ be an $n \times n$ matrix with entries $T_{i j}=\mathbf{c}_{i} \cdot \mathbf{d}_{j}$. Is it the case that $T$ is of maximal rank if and only if the sets of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ and $\left\{\mathbf{d}_{1}, \mathbf{d}_{2}, \ldots, \mathbf{d}_{n}\right\}$ are both linearly independent? Justify your answer with a proof or a counterexample.

  Given an integer $n>2$, is it always possible to find a set of vectors $\left\{\mathbf{c}_{1}, \mathbf{c}_{2}, \ldots, \mathbf{c}_{n}\right\}$ in $\mathbb{R}^{n}$ with the property that every pair is linearly independent and that every triple is linearly dependent? Justify your answer.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2010

  Let $A$ be a complex $n \times n$ matrix with an eigenvalue $\lambda$. Show directly from the definitions that:

  (i) $A^{r}$ has an eigenvalue $\lambda^{r}$ for any integer $r \geqslant 1$; and

  (ii) if $A$ is invertible then $\lambda \neq 0$ and $A^{-1}$ has an eigenvalue $\lambda^{-1}$.

  For any complex $n \times n$ matrix $A$, let $\chi_{A}(t)=\operatorname{det}(A-t I)$. Using standard properties of determinants, show that:

  (iii) $\chi_{A^{2}}\left(t^{2}\right)=\chi_{A}(t) \chi_{A}(-t)$; and

  (iv) if $A$ is invertible,

  $$\chi_{A^{-1}}(t)=(\operatorname{det} A)^{-1}(-1)^{n} t^{n} \chi_{A}\left(t^{-1}\right)$$

  Explain, including justifications, the relationship between the eigenvalues of $A$ and the polynomial $\chi_{A}(t)$.

  If $A^{4}$ has an eigenvalue $\mu$, does it follow that $A$ has an eigenvalue $\lambda$ with $\lambda^{4}=\mu$ ? Give a proof or counterexample.

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• Paper 1, Section II, B

  #Vectors and Matrices

  2010

  Let $R$ be a real orthogonal $3 \times 3$ matrix with a real eigenvalue $\lambda$ corresponding to some real eigenvector. Show algebraically that $\lambda=\pm 1$ and interpret this result geometrically.

  Each of the matrices

  $$M=\left(\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}\right), \quad N=\left(\begin{array}{rrr} 1 & -2 & -2 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right), \quad P=\frac{1}{3}\left(\begin{array}{rrr} 1 & -2 & -2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{array}\right)$$

  has an eigenvalue $\lambda=1$. Confirm this by finding as many independent eigenvectors as possible with this eigenvalue, for each matrix in turn.

  Show that one of the matrices above represents a rotation, and find the axis and angle of rotation. Which of the other matrices represents a reflection, and why?

  State, with brief explanations, whether the matrices $M, N, P$ are diagonalisable (i) over the real numbers; (ii) over the complex numbers.

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• Paper 2, Section I, G

  #Analysis II

  2010

  Let $c>1$ be a real number, and let $F_{c}$ be the space of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers $a_{i}$ with $\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ convergent. Show that $\|\mathbf{a}\|_{c}=\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ defines a norm on $F_{c}$.

  Let $F$ denote the space of sequences a with $\left|a_{i}\right|$ bounded; show that $F \subset F_{c}$. If $c^{\prime}>c$, show that the norms on $F$ given by restricting to $F$ the norms $\|\cdot\|_{c}$ on $F_{c}$ and $\|\cdot\|_{c^{\prime}}$ on $F_{c^{\prime}}$ are not Lipschitz equivalent.

  By considering sequences of the form $\mathbf{a}^{(n)}=\left(a, a^{2}, \ldots, a^{n}, 0,0, \ldots\right)$ in $F$, for $a$ an appropriate real number, or otherwise, show that $F$ (equipped with the norm $\|.\|_{c}$ ) is not complete.

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• Paper 3, Section I, G

  #Analysis II

  2010

  Consider the map $f: \mathbf{R}^{3} \rightarrow \mathbf{R}^{3}$ given by

  $$f(x, y, z)=(x+y+z, x y+y z+z x, x y z)$$

  Show that $f$ is differentiable everywhere and find its derivative.

  Stating carefully any theorem that you quote, show that $f$ is locally invertible near a point $(x, y, z)$ unless $(x-y)(y-z)(z-x)=0$.

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• Paper 4, Section I, G

  #Analysis II

  2010

  Let $S$ denote the set of continuous real-valued functions on the interval $[0,1]$. For $f, g \in S$, set

  $$d_{1}(f, g)=\sup \{|f(x)-g(x)|: x \in[0,1]\} \quad \text { and } \quad d_{2}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$$

  Show that both $d_{1}$ and $d_{2}$ define metrics on $S$. Does the identity map on $S$ define a continuous map of metric spaces $\left(S, d_{1}\right) \rightarrow\left(S, d_{2}\right) ?$ Does the identity map define a continuous map of metric spaces $\left(S, d_{2}\right) \rightarrow\left(S, d_{1}\right)$ ?

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• Paper 1, Section II, G

  #Analysis II

  2010

  State and prove the contraction mapping theorem. Demonstrate its use by showing that the differential equation $f^{\prime}(x)=f\left(x^{2}\right)$, with boundary condition $f(0)=1$, has a unique solution on $[0,1)$, with one-sided derivative $f^{\prime}(0)=1$ at zero.

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• Paper 2, Section II, G

  #Analysis II

  2010

  Suppose the functions $f_{n}(n=1,2, \ldots)$ are defined on the open interval $(0,1)$ and that $f_{n}$ tends uniformly on $(0,1)$ to a function $f$. If the $f_{n}$ are continuous, show that $f$ is continuous. If the $f_{n}$ are differentiable, show by example that $f$ need not be differentiable.

  Assume now that each $f_{n}$ is differentiable and the derivatives $f_{n}^{\prime}$ converge uniformly on $(0,1)$. For any given $c \in(0,1)$, we define functions $g_{c, n}$ by

  $$g_{c, n}(x)= \begin{cases}\frac{f_{n}(x)-f_{n}(c)}{x-c} & \text { for } x \neq c, \\ f_{n}^{\prime}(c) & \text { for } x=c .\end{cases}$$

  Show that each $g_{c, n}$ is continuous. Using the general principle of uniform convergence (the Cauchy criterion) and the Mean Value Theorem, or otherwise, prove that the functions $g_{c, n}$ converge uniformly to a continuous function $g_{c}$ on $(0,1)$, where

  $$g_{c}(x)=\frac{f(x)-f(c)}{x-c} \quad \text { for } x \neq c$$

  Deduce that $f$ is differentiable on $(0,1)$.

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• Paper 3, Section II, G

  #Analysis II

  2010

  Let $f: U \rightarrow \mathbf{R}^{n}$ be a map on an open subset $U \subset \mathbf{R}^{m}$. Explain what it means for $f$ to be differentiable on $U$. If $g: V \rightarrow \mathbf{R}^{m}$ is a differentiable map on an open subset $V \subset \mathbf{R}^{p}$ with $g(V) \subset U$, state and prove the Chain Rule for the derivative of the composite $f g$.

  Suppose now $F: \mathbf{R}^{n} \rightarrow \mathbf{R}$ is a differentiable function for which the partial derivatives $D_{1} F(\mathbf{x})=D_{2} F(\mathbf{x})=\ldots=D_{n} F(\mathbf{x})$ for all $\mathbf{x} \in \mathbf{R}^{n}$. By considering the function $G: \mathbf{R}^{n} \rightarrow \mathbf{R}$ given by

  $$G\left(y_{1}, \ldots, y_{n}\right)=F\left(y_{1}, \ldots, y_{n-1}, y_{n}-\sum_{i=1}^{n-1} y_{i}\right)$$

  or otherwise, show that there exists a differentiable function $h: \mathbf{R} \rightarrow \mathbf{R}$ with $F\left(x_{1}, \ldots, x_{n}\right)=$ $h\left(x_{1}+\cdots+x_{n}\right)$ at all points of $\mathbf{R}^{n} .$

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• Paper 4, Section II, G

  #Analysis II

  2010

  What does it mean to say that a function $f$ on an interval in $\mathbf{R}$ is uniformly continuous? Assuming the Bolzano-Weierstrass theorem, show that any continuous function on a finite closed interval is uniformly continuous.

  Suppose that $f$ is a continuous function on the real line, and that $f(x)$ tends to finite limits as $x \rightarrow \pm \infty$; show that $f$ is uniformly continuous.

  If $f$ is a uniformly continuous function on $\mathbf{R}$, show that $f(x) / x$ is bounded as $x \rightarrow \pm \infty$. If $g$ is a continuous function on $\mathbf{R}$ for which $g(x) / x \rightarrow 0$ as $x \rightarrow \pm \infty$, determine whether $g$ is necessarily uniformly continuous, giving proof or counterexample as appropriate.

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• Paper 4, Section I, G

  #Complex Analysis

  2010

  State the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.

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• Paper 3, Section II, G

  #Complex Analysis

  2010

  State Morera's theorem. Suppose $f_{n}(n=1,2, \ldots)$ are analytic functions on a domain $U \subset \mathbf{C}$ and that $f_{n}$ tends locally uniformly to $f$ on $U$. Show that $f$ is analytic on $U$. Explain briefly why the derivatives $f_{n}^{\prime}$ tend locally uniformly to $f^{\prime}$.

  Suppose now that the $f_{n}$ are nowhere vanishing and $f$ is not identically zero. Let $a$ be any point of $U$; show that there exists a closed disc $\bar{\Delta} \subset U$ with centre $a$, on which the convergence of $f_{n}$ and $f_{n}^{\prime}$ are both uniform, and where $f$ is nowhere zero on $\bar{\Delta} \backslash\{a\}$. By considering

  $$\frac{1}{2 \pi i} \int_{C} \frac{f_{n}^{\prime}(w)}{f_{n}(w)} d w$$

  (where $C$ denotes the boundary of $\bar{\Delta}$ ), or otherwise, deduce that $f(a) \neq 0$.

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• Paper 1, Section II, A

  #Complex Analysis or Complex Methods

  2010

  Calculate the following real integrals by using contour integration. Justify your steps carefully.

  (a)

  $$I_{1}=\int_{0}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x, \quad a>0$$

  (b)

  $$I_{2}=\int_{0}^{\infty} \frac{x^{1 / 2} \log x}{1+x^{2}} d x$$

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• Paper 2, Section II, A

  #Complex Analysis or Complex Methods

  2010

  (a) Prove that a complex differentiable map, $f(z)$, is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of $f(z)$.

  (b) Let $D$ be the region

  $$D:=\{z \in \mathbb{C}:|z-1|>1 \text { and }|z-2|<2\}$$

  Draw the region $D$. It might help to consider the two sets

  $$\begin{aligned} &C(1):=\{z \in \mathbb{C}:|z-1|=1\} \\ &C(2):=\{z \in \mathbb{C}:|z-2|=2\} \end{aligned}$$

  (c) For the transformations below identify the images of $D$.

  Step 1: The first map is $f_{1}(z)=\frac{z-1}{z}$,

  Step 2: The second map is the composite $f_{2} f_{1}$ where $f_{2}(z)=\left(z-\frac{1}{2}\right) i$,

  Step 3: The third map is the composite $f_{3} f_{2} f_{1}$ where $f_{3}(z)=e^{2 \pi z}$.

  (d) Write down the inverse map to the composite $f_{3} f_{2} f_{1}$, explaining any choices of branch.

  [The composite $f_{2} f_{1}$ means $f_{2}\left(f_{1}(z)\right)$.]

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• Paper 3, Section I, A

  #Complex Methods

  2010

  (a) Prove that the real and imaginary parts of a complex differentiable function are harmonic.

  (b) Find the most general harmonic polynomial of the form

  $$u(x, y)=a x^{3}+b x^{2} y+c x y^{2}+d y^{3}$$

  where $a, b, c, d, x$ and $y$ are real.

  (c) Write down a complex analytic function of $z=x+i y$ of which $u(x, y)$ is the real part.

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• Paper 1, Section I, A

  #Complex Analysis or Complex Methods

  2010

  (a) Write down the definition of the complex derivative of the function $f(z)$ of a single complex variable.

  (b) Derive the Cauchy-Riemann equations for the real and imaginary parts $u(x, y)$ and $v(x, y)$ of $f(z)$, where $z=x+i y$ and

  $$f(z)=u(x, y)+i v(x, y)$$

  (c) State necessary and sufficient conditions on $u(x, y)$ and $v(x, y)$ for the function $f(z)$ to be complex differentiable.

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• Paper 4, Section II, A

  #Complex Methods

  2010

  A linear system is described by the differential equation

  $$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=f(t),$$

  with initial conditions

  $$y(0)=0, \quad y^{\prime}(0)=1, \quad y^{\prime \prime}(0)=1$$

  The Laplace transform of $f(t)$ is defined as

  $$\mathcal{L}[f(t)]=\tilde{f}(s)=\int_{0}^{\infty} e^{-s t} f(t) d t$$

  You may assume the following Laplace transforms,

  $$\begin{aligned} \mathcal{L}[y(t)] &=\tilde{y}(s) \\ \mathcal{L}\left[y^{\prime}(t)\right] &=s \tilde{y}(s)-y(0) \\ \mathcal{L}\left[y^{\prime \prime}(t)\right] &=s^{2} \tilde{y}(s)-s y(0)-y^{\prime}(0) \\ \mathcal{L}\left[y^{\prime \prime \prime}(t)\right] &=s^{3} \tilde{y}(s)-s^{2} y(0)-s y^{\prime}(0)-y^{\prime \prime}(0) \end{aligned}$$

  (a) Use Laplace transforms to determine the response, $y_{1}(t)$, of the system to the signal

  $$f(t)=-2$$

  (b) Determine the response, $y_{2}(t)$, given that its Laplace transform is

  $$\tilde{y}_{2}(s)=\frac{1}{s^{2}(s-1)^{2}} .$$

  (c) Given that

  $$y^{\prime \prime \prime}(t)-y^{\prime \prime}(t)-2 y^{\prime}(t)+2 y(t)=g(t)$$

  leads to the response with Laplace transform

  $$\tilde{y}(s)=\frac{1}{s^{2}(s-1)^{2}},$$

  determine $g(t)$.

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• Paper 2, Section I, $\mathbf{6 C}$

  #Electromagnetism

  2010

  Write down Maxwell's equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.

  Show that the homogenous equations (those not involving charge or current densities) can be solved in terms of vector and scalar potentials $\mathbf{A}$ and $\phi$.

  Then re-express the inhomogeneous equations in terms of $\mathbf{A}, \phi$ and $f=\nabla \cdot \mathbf{A}+c^{-2} \dot{\phi}$. Show that the potentials can be chosen so as to set $f=0$ and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation $\nabla^{2} \psi-c^{-2} \ddot{\psi}=\sigma(\mathbf{x}, t)$ always has a solution $\psi(\mathbf{x}, t)$ for any given $\sigma(\mathbf{x}, t)$.]

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• Paper 4, Section I, B

  #Electromagnetism

  2010

  Give an expression for the force $\mathbf{F}$ on a charge $q$ moving at velocity $\mathbf{v}$ in electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$. Consider a stationary electric circuit $C$, and let $S$ be a stationary surface bounded by $C$. Derive from Maxwell's equations the result

  $$\mathcal{E}=-\frac{d \Phi}{d t}$$

  where the electromotive force $\mathcal{E}=\oint_{C} q^{-1} \mathbf{F} \cdot d \mathbf{r}$ and the flux $\Phi=\int_{S} \mathbf{B} \cdot d \mathbf{S}$.

  Now assume that $(*)$ also holds for a moving circuit. A circular loop of wire of radius $a$ and total resistance $R$, whose normal is in the $z$-direction, moves at constant speed $v$ in the $x$-direction in the presence of a magnetic field $\mathbf{B}=\left(0,0, B_{0} x / d\right)$. Find the current in the wire.

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• Paper 1, Section II, C

  #Electromagnetism

  2010

  A capacitor consists of three perfectly conducting coaxial cylinders of radii $a, b$ and $c$ where $0<a<b<c$, and length $L$ where $L \gg c$ so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential $V$. Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.

  Next assume that the radii $a$ and $c$ are fixed, as is the potential $V$, while the radius $b$ is allowed to vary. Show that the energy achieves a minimum when $b$ is the geometric mean of $a$ and $c$.

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• Paper 2, Section II, C

  #Electromagnetism

  2010

  A steady current $I_{2}$ flows around a loop $\mathcal{C}_{2}$ of a perfectly conducting narrow wire. Assuming that the gauge condition $\nabla \cdot \mathbf{A}=0$ holds, the vector potential at points away from the loop may be taken to be

  $$\mathbf{A}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2}}{\left|\mathbf{r}-\mathbf{r}_{2}\right|}$$

  First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field

  $$\mathbf{B}(\mathbf{r})=\frac{\mu_{0} I_{2}}{4 \pi} \oint_{\mathcal{C}_{2}} \frac{d \mathbf{r}_{2} \times\left(\mathbf{r}-\mathbf{r}_{2}\right)}{\left|\mathbf{r}-\mathbf{r}_{2}\right|^{3}}$$

  Next suppose there is a similar but separate loop $\mathcal{C}_{1}$ with current $I_{1}$. Show that the magnetic force exerted on loop $\mathcal{C}_{1}$ by loop $\mathcal{C}_{2}$ is

  $$\mathbf{F}_{12}=\frac{\mu_{0} I_{1} I_{2}}{4 \pi} \oint_{\mathcal{C}_{1}} \oint_{\mathcal{C}_{2}} d \mathbf{r}_{1} \times\left(d \mathbf{r}_{2} \times \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|^{3}}\right)$$

  Is this consistent with Newton's third law? Justify your answer.

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• Paper 3, Section II, C

  #Electromagnetism

  2010

  Write down Maxwell's equations in a region with no charges and no currents. Show that if $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ is a solution then so is $\widetilde{\mathbf{E}}(\mathbf{x}, t)=c \mathbf{B}(\mathbf{x}, t)$ and $\widetilde{\mathbf{B}}(\mathbf{x}, t)=-\mathbf{E}(\mathbf{x}, t) / c$. Write down the boundary conditions on $\mathbf{E}$ and $\mathbf{B}$ at the boundary with unit normal $\mathbf{n}$ between a perfect conductor and a vacuum.

  Electromagnetic waves propagate inside a tube of perfectly conducting material. The tube's axis is in the $z$-direction, and it is surrounded by a vacuum. The fields may be taken to be the real parts of

  $$\mathbf{E}(\mathbf{x}, t)=\mathbf{e}(x, y) e^{i(k z-\omega t)}, \quad \mathbf{B}(\mathbf{x}, t)=\mathbf{b}(x, y) e^{i(k z-\omega t)}$$

  Write down Maxwell's equations in terms of $\mathbf{e}, \mathbf{b}, k$ and $\omega$.

  Suppose first that $b_{z}(x, y)=0$. Show that the solution is determined by

  $$\mathbf{e}=\left(\frac{\partial \psi}{\partial x}, \frac{\partial \psi}{\partial y}, i k\left[1-\frac{\omega^{2}}{k^{2} c^{2}}\right] \psi\right)$$

  where the function $\psi(x, y)$ satisfies

  $$\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}+\gamma^{2} \psi=0$$

  and $\psi$ vanishes on the boundary of the tube. Here $\gamma^{2}$ is a constant whose value should be determined.

  Obtain a similar condition for the case where $e_{z}(x, y)=0$. [You may find it useful to use a result from the first paragraph.] What is the corresponding boundary condition?

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• Paper 1, Section I, B

  #Fluid Dynamics

  2010

  A planar solenoidal velocity field has the velocity potential

  $$\phi(x, y, t)=x e^{-t}+y e^{t}$$

  Find and sketch (i) the streamlines at $t=0$; (ii) the pathline that passes through the origin at $t=0$; (iii) the locus at $t=0$ of points that pass through the origin at earlier times (streakline).

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• Paper 2, Section I, B

  #Fluid Dynamics

  2010

  Write down an expression for the velocity field of a line vortex of strength $\kappa$.

  Consider $N$ identical line vortices of strength $\kappa$ arranged at equal intervals round a circle of radius $a$. Show that the vortices all move around the circle at constant angular velocity $(N-1) \kappa /\left(4 \pi a^{2}\right)$.

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• Paper 1, Section II, B

  #Fluid Dynamics

  2010

  Starting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli's theorem for unsteady irrotational flow.

  Inviscid fluid of density $\rho$ is contained within a U-shaped tube with the arms vertical, of height $h$ and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are $p_{1}$ and $p_{2}$ and the heights of the fluid columns are $\ell_{1}, \ell_{2}$.

  The fluid in arm 1 is displaced upwards by a distance $\xi$ (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli's theorem, show that $\xi(t)$ obeys the equation

  $$\rho\left(\ell_{1}+\ell_{2}\right) \ddot{\xi}+\frac{p_{1} \xi}{h-\ell_{1}-\xi}+\frac{p_{2} \xi}{h-\ell_{2}+\xi}+2 \rho g \xi=0$$

  Now consider the special case $\ell_{1}=\ell_{2}=\ell_{0}, p_{1}=p_{2}=p_{0}$. Construct a first integral of this equation and hence give an expression for the total kinetic energy $\rho \ell_{0} \dot{\xi}^{2}$ of the flow in terms of $\xi$ and the maximum displacement $\xi_{\max }$.

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• Paper 3, Section II, B

  #Fluid Dynamics

  2010

  Write down the exact kinematic and dynamic boundary conditions that apply at the free surface $z=\eta(x, t)$ of a fluid layer in the presence of gravity in the $z$-direction. Show how these may be approximated for small disturbances of a hydrostatic state about $z=0$. (The flow of the fluid is in the $(x, z)$-plane and may be taken to be irrotational, and the pressure at the free surface may be assumed to be constant.)

  Fluid of density $\rho$ fills the region $0>z>-h$. At $z=-h$ the $z$-component of the velocity is $\epsilon \operatorname{Re}\left(e^{i \omega t} \cos k x\right)$, where $|\epsilon| \ll 1$. Find the resulting disturbance of the free surface, assuming this to be small. Explain physically why your answer has a singularity for a particular value of $\omega^{2}$.

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• Paper 4, Section II, B

  #Fluid Dynamics

  2010

  Write down the velocity potential for a line source flow of strength $m$ located at $(r, \theta)=(d, 0)$ in polar coordinates $(r, \theta)$ and derive the velocity components $u_{r}, u_{\theta}$.

  A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius $a(a<d)$ centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at $\left(a^{2} / d, 0\right)$ and another at the origin, of a strength to be determined.

  Use Bernoulli's law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the $x$-direction and of magnitude

  $$\frac{m^{2} \rho}{2 \pi^{2}} \int_{0}^{2 \pi} \frac{a d^{2} \sin ^{2} \theta \cos \theta}{\left(a^{2}+d^{2}-2 a d \cos \theta\right)^{2}} d \theta$$

  where $\rho$ is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.

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• Paper 1, Section I, F

  #Geometry

  2010

  (i) Define the notion of curvature for surfaces embedded in $\mathbb{R}^{3}$.

  (ii) Prove that the unit sphere in $\mathbb{R}^{3}$ has curvature $+1$ at all points.

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• Paper 3, Section I, F

  #Geometry

  2010

  (i) Write down the Poincaré metric on the unit disc model $D$ of the hyperbolic plane. Compute the hyperbolic distance $\rho$ from $(0,0)$ to $(r, 0)$, with $0<r<1$.

  (ii) Given a point $P$ in $D$ and a hyperbolic line $L$ in $D$ with $P$ not on $L$, describe how the minimum distance from $P$ to $L$ is calculated. Justify your answer.

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• Paper 2, Section II, F

  #Geometry

  2010

  Suppose that $a>0$ and that $S \subset \mathbb{R}^{3}$ is the half-cone defined by $z^{2}=a\left(x^{2}+y^{2}\right)$, $z>0$. By using an explicit smooth parametrization of $S$, calculate the curvature of $S$.

  Describe the geodesics on $S$. Show that for $a=3$, no geodesic intersects itself, while for $a>3$ some geodesic does so.

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• Paper 3, Section II, F

  #Geometry

  2010

  Describe the hyperbolic metric on the upper half-plane $H$. Show that any Möbius transformation that preserves $H$ is an isometry of this metric.

  Suppose that $z_{1}, z_{2} \in H$ are distinct and that the hyperbolic line through $z_{1}$ and $z_{2}$ meets the real axis at $w_{1}, w_{2}$. Show that the hyperbolic distance $\rho\left(z_{1}, z_{2}\right)$ between $z_{1}$ and $z_{2}$ is given by $\rho\left(z_{1}, z_{2}\right)=\log r$, where $r$ is the cross-ratio of the four points $z_{1}, z_{2}, w_{1}, w_{2}$, taken in an appropriate order.

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• Paper 4, Section II, F

  #Geometry

  2010

  Suppose that $D$ is the unit disc, with Riemannian metric

  $$d s^{2}=\frac{d x^{2}+d y^{2}}{1-\left(x^{2}+y^{2}\right)}$$

  [Note that this is not a multiple of the Poincaré metric.] Show that the diameters of $D$ are, with appropriate parametrization, geodesics.

  Show that distances between points in $D$ are bounded, but areas of regions in $D$ are unbounded.

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• Paper 2, Section I, $2 \mathrm{H}$

  #Groups, Rings and Modules

  2010

  Give the definition of conjugacy classes in a group $G$. How many conjugacy classes are there in the symmetric group $S_{4}$ on four letters? Briefly justify your answer.

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• Paper 3, Section I, H

  #Groups, Rings and Modules

  2010

  Let $A$ be the ring of integers $\mathbb{Z}$ or the polynomial ring $\mathbb{C}[X]$. In each case, give an example of an ideal $I$ of $A$ such that the quotient ring $R=A / I$ has a non-trivial idempotent (an element $x \in R$ with $x \neq 0,1$ and $x^{2}=x$ ) and a non-trivial nilpotent element (an element $x \in R$ with $x \neq 0$ and $x^{n}=0$ for some positive integer $n$ ). Exhibit these elements and justify your answer.

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• Paper 4, Section I, H

  #Groups, Rings and Modules

  2010

  Let $M$ be a free $\mathbb{Z}$-module generated by $e_{1}$ and $e_{2}$. Let $a, b$ be two non-zero integers, and $N$ be the submodule of $M$ generated by $a e_{1}+b e_{2}$. Prove that the quotient module $M / N$ is free if and only if $a, b$ are coprime.

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• Paper 1, Section II, H

  #Groups, Rings and Modules

  2010

  Prove that the kernel of a group homomorphism $f: G \rightarrow H$ is a normal subgroup of the group $G$.

  Show that the dihedral group $D_{8}$ of order 8 has a non-normal subgroup of order 2. Conclude that, for a group $G$, a normal subgroup of a normal subgroup of $G$ is not necessarily a normal subgroup of $G$.

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• Paper 2, Section II, H

  #Groups, Rings and Modules

  2010

  For ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form $x y$ where $x \in I$ and $y \in J$.

  (1) Prove that, if a prime ideal $P$ of $R$ contains $I J$, then $P$ contains either $I$ or $J$.

  (2) Give an example of $R, I$ and $J$ such that the two ideals $I J$ and $I \cap J$ are different from each other.

  (3) Prove that there is a natural bijection between the prime ideals of $R / I J$ and the prime ideals of $R /(I \cap J)$.

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• Paper 3, Section II, H

  #Groups, Rings and Modules

  2010

  Let $R$ be an integral domain and $R^{\times}$its group of units. An element of $S=R \backslash\left(R^{\times} \cup\{0\}\right)$ is irreducible if it is not a product of two elements in $S$. When $R$ is Noetherian, show that every element of $S$ is a product of finitely many irreducible elements of $S$.

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• Paper 4, Section II, $11 H$

  #Groups, Rings and Modules

  2010

  Let $V=(\mathbb{Z} / 3 \mathbb{Z})^{2}$, a 2-dimensional vector space over the field $\mathbb{Z} / 3 \mathbb{Z}$, and let $e_{1}=\left(\begin{array}{c}1 \\ 0\end{array}\right), e_{2}=\left(\begin{array}{c}0 \\ 1\end{array}\right) \in V .$

  (1) List all 1-dimensional subspaces of $V$ in terms of $e_{1}, e_{2}$. (For example, there is a subspace $\left\langle e_{1}\right\rangle$ generated by $\left.e_{1} .\right)$

  (2) Consider the action of the matrix group

  $$G=G L_{2}(\mathbb{Z} / 3 \mathbb{Z})=\left\{\left(\begin{array}{ll} a & b \\ c & d \end{array}\right) \mid a, b, c, d \in \mathbb{Z} / 3 \mathbb{Z}, \quad a d-b c \neq 0\right\}$$

  on the finite set $X$ of all 1-dimensional subspaces of $V$. Describe the stabiliser group $K$ of $\left\langle e_{1}\right\rangle \in X$. What is the order of $K$ ? What is the order of $G$ ?

  (3) Let $H \subset G$ be the subgroup of all elements of $G$ which act trivially on $X$. Describe $H$, and prove that $G / H$ is isomorphic to $S_{4}$, the symmetric group on four letters.

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• Paper 1, Section I, F

  #Linear Algebra

  2010

  Suppose that $V$ is the complex vector space of polynomials of degree at most $n-1$ in the variable $z$. Find the Jordan normal form for each of the linear transformations $\frac{d}{d z}$ and $z \frac{d}{d z}$ acting on $V$.

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• Paper 2, Section I, F

  #Linear Algebra

  2010

  Suppose that $\phi$ is an endomorphism of a finite-dimensional complex vector space.

  (i) Show that if $\lambda$ is an eigenvalue of $\phi$, then $\lambda^{2}$ is an eigenvalue of $\phi^{2}$.

  (ii) Show conversely that if $\mu$ is an eigenvalue of $\phi^{2}$, then there is an eigenvalue $\lambda$ of $\phi$ with $\lambda^{2}=\mu$.

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• Paper 4, Section I, F

  #Linear Algebra

  2010

  Define the notion of an inner product on a finite-dimensional real vector space $V$, and the notion of a self-adjoint linear map $\alpha: V \rightarrow V$.

  Suppose that $V$ is the space of real polynomials of degree at most $n$ in a variable $t$. Show that

  $$\langle f, g\rangle=\int_{-1}^{1} f(t) g(t) d t$$

  is an inner product on $V$, and that the map $\alpha: V \rightarrow V$ :

  $$\alpha(f)(t)=\left(1-t^{2}\right) f^{\prime \prime}(t)-2 t f^{\prime}(t)$$

  is self-adjoint.

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• Paper 1, Section II, F

  #Linear Algebra

  2010

  Let $V$ denote the vector space of $n \times n$ real matrices.

  (1) Show that if $\psi(A, B)=\operatorname{tr}\left(A B^{T}\right)$, then $\psi$ is a positive-definite symmetric bilinear form on $V$.

  (2) Show that if $q(A)=\operatorname{tr}\left(A^{2}\right)$, then $q$ is a quadratic form on $V$. Find its rank and signature.

  [Hint: Consider symmetric and skew-symmetric matrices.]

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• Paper 2, Section II, F

  #Linear Algebra

  2010

  (i) Show that two $n \times n$ complex matrices $A, B$ are similar (i.e. there exists invertible $P$ with $A=P^{-1} B P$ ) if and only if they represent the same linear map $\mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ with respect to different bases.

  (ii) Explain the notion of Jordan normal form of a square complex matrix.

  (iii) Show that any square complex matrix $A$ is similar to its transpose.

  (iv) If $A$ is invertible, describe the Jordan normal form of $A^{-1}$ in terms of that of $A$.

  Justify your answers.

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• Paper 3, Section II, F

  #Linear Algebra

  2010

  Suppose that $V$ is a finite-dimensional vector space over $\mathbb{C}$, and that $\alpha: V \rightarrow V$ is a $\mathbb{C}$-linear map such that $\alpha^{n}=1$ for some $n>1$. Show that if $V_{1}$ is a subspace of $V$ such that $\alpha\left(V_{1}\right) \subset V_{1}$, then there is a subspace $V_{2}$ of $V$ such that $V=V_{1} \oplus V_{2}$ and $\alpha\left(V_{2}\right) \subset V_{2}$.

  [Hint: Show, for example by picking bases, that there is a linear map $\pi: V \rightarrow V_{1}$ with $\pi(x)=x$ for all $x \in V_{1}$. Then consider $\rho: V \rightarrow V_{1}$ with $\left.\rho(y)=\frac{1}{n} \sum_{i=0}^{n-1} \alpha^{i} \pi \alpha^{-i}(y) .\right]$

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• Paper 4, Section II, F

  #Linear Algebra

  2010

  (i) Show that the group $O_{n}(\mathbb{R})$ of orthogonal $n \times n$ real matrices has a normal subgroup $S O_{n}(\mathbb{R})=\left\{A \in O_{n}(\mathbb{R}) \mid \operatorname{det} A=1\right\}$.

  (ii) Show that $O_{n}(\mathbb{R})=S O_{n}(\mathbb{R}) \times\left\{\pm I_{n}\right\}$ if and only if $n$ is odd.

  (iii) Show that if $n$ is even, then $O_{n}(\mathbb{R})$ is not the direct product of $S O_{n}(\mathbb{R})$ with any normal subgroup.

  [You may assume that the only elements of $O_{n}(\mathbb{R})$ that commute with all elements of $O_{n}(\mathbb{R})$ are $\pm I_{n}$.]

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• Paper 3, Section I, E

  #Markov Chains

  2010

  An intrepid tourist tries to ascend Springfield's famous infinite staircase on an icy day. When he takes a step with his right foot, he reaches the next stair with probability $1 / 2$, otherwise he falls down and instantly slides back to the bottom with probability $1 / 2$. Similarly, when he steps with his left foot, he reaches the next stair with probability $1 / 3$, or slides to the bottom with probability $2 / 3$. Assume that he always steps first with his right foot when he is at the bottom, and alternates feet as he ascends. Let $X_{n}$ be his position after his $n$th step, so that $X_{n}=i$ when he is on the stair $i, i=0,1,2, \ldots$, where 0 is the bottom stair.

  (a) Specify the transition probabilities $p_{i j}$ for the Markov chain $\left(X_{n}\right)_{n} \geqslant 0$ for any $i, j \geqslant 0$.

  (b) Find the equilibrium probabilities $\pi_{i}$, for $i \geqslant 0$. [Hint: $\left.\pi_{0}=5 / 9 .\right]$

  (c) Argue that the chain is irreducible and aperiodic and evaluate the limit

  $$\lim _{n \rightarrow \infty} \mathbb{P}\left(X_{n}=i\right)$$

  for each $i \geqslant 0$.

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• Paper 4, Section I, E

  #Markov Chains

  2010

  Consider a Markov chain $\left(X_{n}\right)_{n} \geqslant 0$ with state space $\{a, b, c, d\}$ and transition probabilities given by the following table.

  \begin{tabular}{c|cccc} & $a$ & $b$ & $c$ & $d$ \ \hline$a$ & $1 / 4$ & $1 / 4$ & $1 / 2$ & 0 \ $b$ & 0 & $1 / 4$ & 0 & $3 / 4$ \ $c$ & $1 / 2$ & 0 & $1 / 4$ & $1 / 4$ \ $d$ & 0 & $1 / 2$ & 0 & $1 / 2$ \end{tabular}

  By drawing an appropriate diagram, determine the communicating classes of the chain, and classify them as either open or closed. Compute the following transition and hitting probabilities:

  • $\mathbb{P}\left(X_{n}=b \mid X_{0}=d\right)$ for a fixed $n \geqslant 0$

  • $\mathbb{P}\left(X_{n}=c\right.$ for some $\left.n \geqslant 1 \mid X_{0}=a\right)$.

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• Paper 1, Section II, E

  #Markov Chains

  2010

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain.

  (a) What does it mean to say that a state $i$ is positive recurrent? How is this property related to the equilibrium probability $\pi_{i}$ ? You do not need to give a full proof, but you should carefully state any theorems you use.

  (b) What is a communicating class? Prove that if states $i$ and $j$ are in the same communicating class and $i$ is positive recurrent then $j$ is positive recurrent also.

  A frog is in a pond with an infinite number of lily pads, numbered $1,2,3, \ldots$ She hops from pad to pad in the following manner: if she happens to be on pad $i$ at a given time, she hops to one of pads $(1,2, \ldots, i, i+1)$ with equal probability.

  (c) Find the equilibrium distribution of the corresponding Markov chain.

  (d) Now suppose the frog starts on pad $k$ and stops when she returns to it. Show that the expected number of times the frog hops is $e(k-1)$ ! where $e=2.718 \ldots$ What is the expected number of times she will visit the lily pad $k+1$ ?

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• Paper 2, Section II, E

  #Markov Chains

  2010

  Let $\left(X_{n}\right)_{n \geqslant 0}$ be a simple, symmetric random walk on the integers $\{\ldots,-1,0,1, \ldots\}$, with $X_{0}=0$ and $\mathbb{P}\left(X_{n+1}=i \pm 1 \mid X_{n}=i\right)=1 / 2$. For each integer $a \geqslant 1$, let $T_{a}=\inf \left\{n \geqslant 0: X_{n}=a\right\}$. Show that $T_{a}$ is a stopping time.

  Define a random variable $Y_{n}$ by the rule

  $$Y_{n}= \begin{cases}X_{n} & \text { if } n<T_{a} \\ 2 a-X_{n} & \text { if } n \geqslant T_{a}\end{cases}$$

  Show that $\left(Y_{n}\right)_{n} \geqslant 0$ is also a simple, symmetric random walk.

  Let $M_{n}=\max _{0 \leqslant i \leqslant n} X_{n}$. Explain why $\left\{M_{n} \geqslant a\right\}=\left\{T_{a} \leqslant n\right\}$ for $a \geqslant 0$. By using the process $\left(Y_{n}\right)_{n \geqslant 0}$ constructed above, show that, for $a \geqslant 0$,

  $$\mathbb{P}\left(M_{n} \geqslant a, X_{n} \leqslant a-1\right)=\mathbb{P}\left(X_{n} \geqslant a+1\right),$$

  and thus

  $$\mathbb{P}\left(M_{n} \geqslant a\right)=\mathbb{P}\left(X_{n} \geqslant a\right)+\mathbb{P}\left(X_{n} \geqslant a+1\right)$$

  Hence compute

  $$\mathbb{P}\left(M_{n}=a\right)$$

  when $a$ and $n$ are positive integers with $n \geqslant a$. [Hint: if $n$ is even, then $X_{n}$ must be even, and if $n$ is odd, then $X_{n}$ must be odd.]

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• Paper 2, Section I, A

  #Methods

  2010

  Consider the initial value problem

  $$\mathcal{L} x(t)=f(t), \quad x(0)=0, \quad \dot{x}(0)=0, \quad t \geqslant 0,$$

  where $\mathcal{L}$ is a second-order linear operator involving differentiation with respect to $t$. Explain briefly how to solve this by using a Green's function.

  Now consider

  $$\ddot{x}(t)= \begin{cases}a & 0 \leqslant t \leqslant T \\ 0 & T<t<\infty\end{cases}$$

  where $a$ is a constant, subject to the same initial conditions. Solve this using the Green's function, and explain how your answer is related to a problem in Newtonian dynamics.

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• Paper 3, Section I, B

  #Methods

  2010

  Show that Laplace's equation $\nabla^{2} \phi=0$ in polar coordinates $(r, \theta)$ has solutions proportional to $r^{\pm \alpha} \sin \alpha \theta, r^{\pm \alpha} \cos \alpha \theta$ for any constant $\alpha$.

  Find the function $\phi$ satisfying Laplace's equation in the region $a<r<b, 0<\theta<\pi$, where $\phi(a, \theta)=\sin ^{3} \theta, \phi(b, \theta)=\phi(r, 0)=\phi(r, \pi)=0$.

  [The Laplacian $\nabla^{2}$ in polar coordinates is

  $$\left.\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}\right]$$

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• Paper 4, Section I, A

  #Methods

  2010

  (a) By considering strictly monotonic differentiable functions $\varphi(x)$, such that the zeros satisfy $\varphi(c)=0$ but $\varphi^{\prime}(c) \neq 0$, establish the formula

  $$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$$

  Hence show that for a general differentiable function with only such zeros, labelled by $c$,

  $$\int_{-\infty}^{\infty} f(x) \delta(\varphi(x)) d x=\sum_{c} \frac{f(c)}{\left|\varphi^{\prime}(c)\right|}$$

  (b) Hence by changing to plane polar coordinates, or otherwise, evaluate,

  $$I=\int_{0}^{\infty} \int_{0}^{\infty}\left(x^{3}+y^{2} x\right) \delta\left(x^{2}+y^{2}-1\right) d y d x$$

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• Paper 1, Section II, A

  #Methods

  2010

  (a) A function $f(t)$ is periodic with period $2 \pi$ and has continuous derivatives up to and including the $k$ th derivative. Show by integrating by parts that the Fourier coefficients of $f(t)$

  $$\begin{aligned} &a_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \cos n t d t \\ &b_{n}=\frac{1}{\pi} \int_{0}^{2 \pi} f(t) \sin n t d t \end{aligned}$$

  decay at least as fast as $1 / n^{k}$ as $n \rightarrow \infty$

  (b) Calculate the Fourier series of $f(t)=|\sin t|$ on $[0,2 \pi]$.

  (c) Comment on the decay rate of your Fourier series.

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• Paper 2, Section II, B

  #Methods

  2010

  Explain briefly the use of the method of characteristics to solve linear first-order partial differential equations.

  Use the method to solve the problem

  $$(x-y) \frac{\partial u}{\partial x}+(x+y) \frac{\partial u}{\partial y}=\alpha u$$

  where $\alpha$ is a constant, with initial condition $u(x, 0)=x^{2}, x \geqslant 0$.

  By considering your solution explain:

  (i) why initial conditions cannot be specified on the whole $x$-axis;

  (ii) why a single-valued solution in the entire plane is not possible if $\alpha \neq 2$.

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• Paper 3, Section II, A

  #Methods

  2010

  (a) Put the equation

  $$x \frac{d^{2} u}{d x^{2}}+\frac{d u}{d x}+\lambda x u=0, \quad 0 \leqslant x \leqslant 1$$

  into Sturm-Liouville form.

  (b) Suppose $u_{n}(x)$ are eigenfunctions such that $u_{n}(x)$ are bounded as $x$ tends to zero and

  $$x \frac{d^{2} u_{n}}{d x^{2}}+\frac{d u_{n}}{d x}+\lambda_{n} x u_{n}=0, \quad 0 \leqslant x \leqslant 1$$

  Identify the weight function $w(x)$ and the most general boundary conditions on $u_{n}(x)$ which give the orthogonality relation

  $$\left(\lambda_{m}-\lambda_{n}\right) \int_{0}^{1} u_{m}(x) w(x) u_{n}(x) d x=0$$

  (c) The equation

  $$x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}+x y=0, \quad x>0$$

  has a solution $J_{0}(x)$ and a second solution which is not bounded at the origin. The zeros of $J_{0}(x)$ arranged in ascending order are $j_{n}, n=1,2, \ldots$. Given that $u_{n}(1)=0$, show that the eigenvalues of the Sturm-Liouville problem in (b) are $\lambda=j_{n}^{2}, n=1,2, \ldots$

  (d) Using the differential equations for $J_{0}(\alpha x)$ and $J_{0}(\beta x)$ and integration by parts, show that

  $$\int_{0}^{1} J_{0}(\alpha x) J_{0}(\beta x) x d x=\frac{\beta J_{0}(\alpha) J_{0}^{\prime}(\beta)-\alpha J_{0}(\beta) J_{0}^{\prime}(\alpha)}{\alpha^{2}-\beta^{2}} \quad(\alpha \neq \beta)$$

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• Paper 4, Section II, B

  #Methods

  2010

  Defining the function $G_{f_{3}}\left(\mathbf{r} ; \mathbf{r}_{0}\right)=-1 /\left(4 \pi\left|\mathbf{r}-\mathbf{r}_{0}\right|\right)$, prove Green's third identity for functions $u(\mathbf{r})$ satisfying Laplace's equation in a volume $V$ with surface $S$, namely

  $$u\left(\mathbf{r}_{0}\right)=\int_{S}\left(u \frac{\partial G_{f_{3}}}{\partial n}-\frac{\partial u}{\partial n} G_{f_{3}}\right) d S$$

  A solution is sought to the Neumann problem for $\nabla^{2} u=0$ in the half plane $z>0$ :

  $$u=O\left(|\mathbf{x}|^{-a}\right), \quad \frac{\partial u}{\partial r}=O\left(|\mathbf{x}|^{-a-1}\right) \text { as }|\mathbf{x}| \rightarrow \infty, \quad \frac{\partial u}{\partial z}=p(x, y) \text { on } z=0$$

  where $a>0$. It is assumed that $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) d x d y=0$. Explain why this condition is necessary.

  Construct an appropriate Green's function $G\left(\mathbf{r} ; \mathbf{r}_{0}\right)$ satisfying $\partial G / \partial z=0$ at $z=0$, using the method of images or otherwise. Hence find the solution in the form

  $$u\left(x_{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) f\left(x-x_{0}, y-y_{0}, z_{0}\right) d x d y$$

  where $f$ is to be determined.

  Now let

  $$p(x, y)= \begin{cases}x & |x|,|y|<a \\ 0 & \text { otherwise }\end{cases}$$

  By expanding $f$ in inverse powers of $z_{0}$, show that

  $$u \rightarrow \frac{-2 a^{4} x_{0}}{3 \pi z_{0}^{3}} \quad \text { as } \quad z_{0} \rightarrow \infty .$$

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• Paper 2, Section I, H

  #Metric and Topological Spaces

  2010

  On the set $\mathbb{Q}$ of rational numbers, the 3 -adic metric $d_{3}$ is defined as follows: for $x, y \in \mathbb{Q}$, define $d_{3}(x, x)=0$ and $d_{3}(x, y)=3^{-n}$, where $n$ is the integer satisfying $x-y=3^{n} u$ where $u$ is a rational number whose denominator and numerator are both prime to 3 .

  (1) Show that this is indeed a metric on $\mathbb{Q}$.

  (2) Show that in $\left(\mathbb{Q}, d_{3}\right)$, we have $3^{n} \rightarrow 0$ as $n \rightarrow \infty$ while $3^{-n} \nrightarrow \infty$ as $n \rightarrow \infty$. Let $d$ be the usual metric $d(x, y)=|x-y|$ on $\mathbb{Q}$. Show that neither the identity map $(\mathbb{Q}, d) \rightarrow\left(\mathbb{Q}, d_{3}\right)$ nor its inverse is continuous.

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• Paper 3, Section I, H

  #Metric and Topological Spaces

  2010

  Let $X$ be a topological space and $Y$ be a set. Let $p: X \rightarrow Y$ be a surjection. The quotient topology on $Y$ is defined as follows: a subset $V \subset Y$ is open if and only if $p^{-1}(V)$ is open in $X$.

  (1) Show that this does indeed define a topology on $Y$, and show that $p$ is continuous when we endow $Y$ with this topology.

  (2) Let $Z$ be another topological space and $f: Y \rightarrow Z$ be a map. Show that $f$ is continuous if and only if $f \circ p: X \rightarrow Z$ is continuous.

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• Paper 1, Section II, H

  #Metric and Topological Spaces

  2010

  Let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ be continuous maps of topological spaces with $f \circ g=\mathrm{id}_{Y}$.

  (1) Suppose that (i) $Y$ is path-connected, and (ii) for every $y \in Y$, its inverse image $f^{-1}(y)$ is path-connected. Prove that $X$ is path-connected.

  (2) Prove the same statement when "path-connected" is everywhere replaced by "connected".

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• Paper 4, Section II, H

  #Metric and Topological Spaces

  2010

  (1) Prove that if $X$ is a compact topological space, then a closed subset $Y$ of $X$ endowed with the subspace topology is compact.

  (2) Consider the following equivalence relation on $\mathbb{R}^{2}$ :

  $$\left(x_{1}, y_{1}\right) \sim\left(x_{2}, y_{2}\right) \Longleftrightarrow\left(x_{1}-x_{2}, y_{1}-y_{2}\right) \in \mathbb{Z}^{2}$$

  Let $X=\mathbb{R}^{2} / \sim$ be the quotient space endowed with the quotient topology, and let $p: \mathbb{R}^{2} \rightarrow X$ be the canonical surjection mapping each element to its equivalence class. Let $Z=\left\{(x, y) \in \mathbb{R}^{2} \mid y=\sqrt{2} x\right\} .$

  (i) Show that $X$ is compact.

  (ii) Assuming that $p(Z)$ is dense in $X$, show that $\left.p\right|_{Z}: Z \rightarrow p(Z)$ is bijective but not homeomorphic.

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• Paper 1, Section I, C

  #Numerical Analysis

  2010

  Obtain the Cholesky decompositions of

  $$H_{3}=\left(\begin{array}{ccc} 1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \end{array}\right), \quad H_{4}=\left(\begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \lambda \end{array}\right) .$$

  What is the minimum value of $\lambda$ for $H_{4}$ to be positive definite? Verify that if $\lambda=\frac{1}{7}$ then $H_{4}$ is positive definite.

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• Paper 4, Section I, C

  #Numerical Analysis

  2010

  Suppose $x_{0}, x_{1}, \ldots, x_{n} \in[a, b] \subset \mathbf{R}$ are pointwise distinct and $f(x)$ is continuous on $[a, b]$. For $k=1,2, \ldots, n$ define

  $$I_{0, k}(x)=\frac{f\left(x_{0}\right)\left(x_{k}-x\right)-f\left(x_{k}\right)\left(x_{0}-x\right)}{x_{k}-x_{0}}$$

  and for $k=2,3, \ldots, n$

  $$I_{0,1, \ldots, k-2, k-1, k}(x)=\frac{I_{0,1, \ldots, k-2, k-1}(x)\left(x_{k}-x\right)-I_{0,1, \ldots, k-2, k}(x)\left(x_{k-1}-x\right)}{x_{k}-x_{k-1}}$$

  Show that $I_{0,1}, \ldots, k-2, k-1, k(x)$ is a polynomial of order $k$ which interpolates $f(x)$ at $x_{0}, x_{1}, \ldots, x_{k}$.

  Given $x_{k}=\{-1,0,2,5\}$ and $f\left(x_{k}\right)=\{33,5,9,1335\}$, determine the interpolating polynomial.

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• Paper 1, Section II, 18C

  #Numerical Analysis

  2010

  Let

  $$\langle f, g\rangle=\int_{-\infty}^{\infty} e^{-x^{2}} f(x) g(x) d x$$

  be an inner product. The Hermite polynomials $H_{n}(x), n=0,1,2, \ldots$ are polynomials in $x$ of degree $n$ with leading term $2^{n} x^{n}$ which are orthogonal with respect to the inner product, with

  $$\left\langle H_{m}, H_{n}\right\rangle= \begin{cases}\gamma_{m}>0 & \text { if } m=n \\ 0 & \text { otherwise }\end{cases}$$

  and $H_{0}(x)=1$. Find a three-term recurrence relation which is satisfied by $H_{n}(x)$ and $\gamma_{n}$ for $n=1,2,3$. [You may assume without proof that

  $$\left.\langle 1,1\rangle=\sqrt{\pi}, \quad\langle x, x\rangle=\frac{1}{2} \sqrt{\pi}, \quad\left\langle x^{2}, x^{2}\right\rangle=\frac{3}{4} \sqrt{\pi}, \quad\left\langle x^{3}, x^{3}\right\rangle=\frac{15}{8} \sqrt{\pi} .\right]$$

  Next let $x_{0}, x_{1}, \ldots, x_{k}$ be the $k+1$ distinct zeros of $H_{k+1}(x)$ and for $i, j=0,1, \ldots, k$ define the Lagrangian polynomials

  $$L_{i}(x)=\prod_{j \neq i} \frac{x-x_{j}}{x_{i}-x_{j}}$$

  associated with these points. Prove that $\left\langle L_{i}, L_{j}\right\rangle=0$ if $i \neq j$.

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• Paper 2, Section II, C

  #Numerical Analysis

  2010

  Consider the initial value problem for an autonomous differential equation

  $$y^{\prime}(t)=f(y(t)), \quad y(0)=y_{0} \text { given }$$

  and its approximation on a grid of points $t_{n}=n h, n=0,1,2, \ldots$. Writing $y_{n}=y\left(t_{n}\right)$, it is proposed to use one of two Runge-Kutta schemes defined by

  $$y_{n+1}=y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)$$

  where $k_{1}=h f\left(y_{n}\right)$ and

  $$k_{2}= \begin{cases}h f\left(y_{n}+k_{1}\right) & \text { scheme I } \\ h f\left(y_{n}+\frac{1}{2}\left(k_{1}+k_{2}\right)\right) & \text { scheme II }\end{cases}$$

  What is the order of each scheme? Determine the $A$-stability of each scheme.

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• Paper 3, Section II, C

  #Numerical Analysis

  2010

  Define the QR factorization of an $m \times n$ matrix $A$ and explain how it can be used to solve the least squares problem of finding the $x^{*} \in \mathbf{R}^{n}$ which minimises $\|A x-b\|$ where $b \in \mathbf{R}^{m}, m>n$, and the norm is the Euclidean one.

  Define a Householder (reflection) transformation $H$ and show that it is an orthogonal matrix.

  Using a Householder reflection, solve the least squares problem for

  $$A=\left(\begin{array}{rrr} 2 & 4 & 7 \\ 0 & 3 & -1 \\ 0 & 0 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & -2 \end{array}\right), \quad b=\left(\begin{array}{r} 9 \\ -7 \\ 3 \\ 1 \\ -1 \end{array}\right)$$

  giving both $x^{*}$ and $\left\|A x^{*}-b\right\|$.

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• Paper 1, Section I, 8E

  #Optimization

  2010

  What is the maximal flow problem in a network?

  Explain the Ford-Fulkerson algorithm. Why must this algorithm terminate if the initial flow is set to zero and all arc capacities are rational numbers?

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• Paper 2, Section I, E

  #Optimization

  2010

  Consider the function $\phi$ defined by

  $$\phi(b)=\inf \left\{x^{2}+y^{4}: x+2 y=b\right\}$$

  Use the Lagrangian sufficiency theorem to evaluate $\phi(3)$. Compute the derivative $\phi^{\prime}(3)$.

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• Paper 3 , Section II, E

  #Optimization

  2010

  Let $A$ be the $m \times n$ payoff matrix of a two-person, zero-sum game. What is Player I's optimization problem?

  Write down a sufficient condition that a vector $p \in \mathbb{R}^{m}$ is an optimal mixed strategy for Player I in terms of the optimal mixed strategy of Player II and the value of the game. If $m=n$ and $A$ is an invertible, symmetric matrix such that $A^{-1} e \geqslant 0$, where $e=(1, \ldots, 1)^{T} \in \mathbb{R}^{m}$, show that the value of the game is $\left(e^{T} A^{-1} e\right)^{-1}$

  Consider the following game: Players I and II each have three cards labelled 1,2 , and 3. Each player chooses one of her cards, independently of the other, and places it in the same envelope. If the sum of the numbers in the envelope is smaller than or equal to 4, then Player II pays Player I the sum (in $£$ ), and otherwise Player I pays Player II the sum. (For instance, if Player I chooses card 3 and Player II choose card 2, then Player I pays Player II £5.) What is the optimal strategy for each player?

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• Paper 4, Section II, E

  #Optimization

  2010

  A factory produces three types of sugar, types $X$, $Y$, and $Z$, from three types of syrup, labelled $A, B$, and C. The following table contains the number of litres of syrup necessary to make each kilogram of sugar.

  \begin{tabular}{c|ccc} & $\mathrm{X}$ & $\mathrm{Y}$ & $\mathrm{Z}$ \ \hline $\mathrm{A}$ & 3 & 2 & 1 \ $\mathrm{~B}$ & 2 & 3 & 2 \ $\mathrm{C}$ & 4 & 1 & 2 \end{tabular}

  For instance, one kilogram of type $\mathrm{X}$ sugar requires 3 litres of $\mathrm{A}, 2$ litres of $\mathrm{B}$, and 4 litres of C. The factory can sell each type of sugar for one pound per kilogram. Assume that the factory owner can use no more than 44 litres of $\mathrm{A}$ and 51 litres of $\mathrm{B}$, but is required by law to use at least 12 litres of C. If her goal is to maximize profit, how many kilograms of each type of sugar should the factory produce?

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• Paper 3, Section I, D

  #Quantum Mechanics

  2010

  Write down the commutation relations between the components of position $\mathbf{x}$ and momentum $\mathbf{p}$ for a particle in three dimensions.

  A particle of mass $m$ executes simple harmonic motion with Hamiltonian

  $$H=\frac{1}{2 m} \mathbf{p}^{2}+\frac{m \omega^{2}}{2} \mathbf{x}^{2},$$

  and the orbital angular momentum operator is defined by

  $$\mathbf{L}=\mathbf{x} \times \mathbf{p}$$

  Show that the components of $\mathbf{L}$ are observables commuting with $H$. Explain briefly why the components of $\mathbf{L}$ are not simultaneous observables. What are the implications for the labelling of states of the three-dimensional harmonic oscillator?

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• Paper 4, Section I, D

  #Quantum Mechanics

  2010

  Determine the possible values of the energy of a particle free to move inside a cube of side $a$, confined there by a potential which is infinite outside and zero inside.

  What is the degeneracy of the lowest-but-one energy level?

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• Paper 1, Section II, 15D

  #Quantum Mechanics

  2010

  A particle of unit mass moves in one dimension in a potential

  $$V=\frac{1}{2} \omega^{2} x^{2}$$

  Show that the stationary solutions can be written in the form

  $$\psi_{n}(x)=f_{n}(x) \exp \left(-\alpha x^{2}\right)$$

  You should give the value of $\alpha$ and derive any restrictions on $f_{n}(x)$. Hence determine the possible energy eigenvalues $E_{n}$.

  The particle has a wave function $\psi(x, t)$ which is even in $x$ at $t=0$. Write down the general form for $\psi(x, 0)$, using the fact that $f_{n}(x)$ is an even function of $x$ only if $n$ is even. Hence write down $\psi(x, t)$ and show that its probability density is periodic in time with period $\pi / \omega$.

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• Paper 2, Section II, D

  #Quantum Mechanics

  2010

  A particle of mass $m$ moves in a one-dimensional potential defined by

  $$V(x)= \begin{cases}\infty & \text { for } x<0 \\ 0 & \text { for } 0 \leqslant x \leqslant a \\ V_{0} & \text { for } a<x\end{cases}$$

  where $a$ and $V_{0}$ are positive constants. Defining $c=\left[2 m\left(V_{0}-E\right)\right]^{1 / 2} / \hbar$ and $k=$ $(2 m E)^{1 / 2} / \hbar$, show that for any allowed positive value $E$ of the energy with $E<V_{0}$ then

  $$c+k \cot k a=0$$

  Find the minimum value of $V_{0}$ for this equation to have a solution.

  Find the normalized wave function for the particle. Write down an expression for the expectation value of $x$ in terms of two integrals, which you need not evaluate. Given that

  $$\langle x\rangle=\frac{1}{2 k}(k a-\tan k a),$$

  discuss briefly the possibility of $\langle x\rangle$ being greater than $a$. [Hint: consider the graph of - ka cot $k a$ against $k a .]$

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• Paper 3, Section II, D

  #Quantum Mechanics

  2010

  A $\pi^{-}$(a particle of the same charge as the electron but 270 times more massive) is bound in the Coulomb potential of a proton. Assuming that the wave function has the form $c e^{-r / a}$, where $c$ and $a$ are constants, determine the normalized wave function of the lowest energy state of the $\pi^{-}$, assuming it to be an $S$-wave (i.e. the state with $l=0$ ). (You should treat the proton as fixed in space.)

  Calculate the probability of finding the $\pi^{-}$inside a sphere of radius $R$ in terms of the ratio $\mu=R / a$, and show that this probability is given by $4 \mu^{3} / 3+O\left(\mu^{4}\right)$ if $\mu$ is very small. Would the result be larger or smaller if the $\pi^{-}$were in a $P$-wave $(l=1)$ state? Justify your answer very briefly.

  [Hint: in spherical polar coordinates,

  $$\left.\nabla^{2} \psi(\mathbf{r})=\frac{1}{r} \frac{\partial^{2}}{\partial r^{2}}(r \psi)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \psi}{\partial \phi^{2}}\right]$$

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• Paper 1, Section I, E

  #Statistics

  2010

  Suppose $X_{1}, \ldots, X_{n}$ are independent $N\left(0, \sigma^{2}\right)$ random variables, where $\sigma^{2}$ is an unknown parameter. Explain carefully how to construct the uniformly most powerful test of size $\alpha$ for the hypothesis $H_{0}: \sigma^{2}=1$ versus the alternative $H_{1}: \sigma^{2}>1$.

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• Paper 2, Section I, E

  #Statistics

  2010

  A washing powder manufacturer wants to determine the effectiveness of a television advertisement. Before the advertisement is shown, a pollster asks 100 randomly chosen people which of the three most popular washing powders, labelled $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$, they prefer. After the advertisement is shown, another 100 randomly chosen people (not the same as before) are asked the same question. The results are summarized below.

  \begin{tabular}{c|ccc} & $\mathrm{A}$ & $\mathrm{B}$ & $\mathrm{C}$ \ \hline before & 36 & 47 & 17 \ after & 44 & 33 & 23 \end{tabular}

  Derive and carry out an appropriate test at the $5 \%$ significance level of the hypothesis that the advertisement has had no effect on people's preferences.

  [You may find the following table helpful:

  $\left.\begin{array}{c|cccccc} & \chi_{1}^{2} & \chi_{2}^{2} & \chi_{3}^{2} & \chi_{4}^{2} & \chi_{5}^{2} & \chi_{6}^{2} \\ \hline 95 \text { percentile } & 3.84 & 5.99 & 7.82 & 9.49 & 11.07 & 12.59\end{array} \cdot\right]$

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• Paper 1, Section II, E

  #Statistics

  2010

  Consider the the linear regression model

  $$Y_{i}=\beta x_{i}+\epsilon_{i},$$

  where the numbers $x_{1}, \ldots, x_{n}$ are known, the independent random variables $\epsilon_{1}, \ldots, \epsilon_{n}$ have the $N\left(0, \sigma^{2}\right)$ distribution, and the parameters $\beta$ and $\sigma^{2}$ are unknown. Find the maximum likelihood estimator for $\beta$.

  State and prove the Gauss-Markov theorem in the context of this model.

  Write down the distribution of an arbitrary linear estimator for $\beta$. Hence show that there exists a linear, unbiased estimator $\widehat{\beta}$ for $\beta$ such that

  $$\mathbb{E}_{\beta, \sigma^{2}}\left[(\widehat{\beta}-\beta)^{4}\right] \leqslant \mathbb{E}_{\beta, \sigma^{2}}\left[(\widetilde{\beta}-\beta)^{4}\right]$$

  for all linear, unbiased estimators $\widetilde{\beta}$.

  [Hint: If $Z \sim N\left(a, b^{2}\right)$ then $\left.\mathbb{E}\left[(Z-a)^{4}\right]=3 b^{4} .\right]$

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• Paper 3, Section II, E

  #Statistics

  2010

  Let $X_{1}, \ldots, X_{n}$ be independent $\operatorname{Exp}(\theta)$ random variables with unknown parameter $\theta$. Find the maximum likelihood estimator $\hat{\theta}$ of $\theta$, and state the distribution of $n / \hat{\theta}$. Show that $\theta / \hat{\theta}$ has the $\Gamma(n, n)$ distribution. Find the $100(1-\alpha) \%$ confidence interval for $\theta$ of the form $[0, C \hat{\theta}]$ for a constant $C>0$ depending on $\alpha$.

  Now, taking a Bayesian point of view, suppose your prior distribution for the parameter $\theta$ is $\Gamma(k, \lambda)$. Show that your Bayesian point estimator $\hat{\theta}_{B}$ of $\theta$ for the loss function $L(\theta, a)=(\theta-a)^{2}$ is given by

  $$\hat{\theta}_{B}=\frac{n+k}{\lambda+\sum_{i} X_{i}} .$$

  Find a constant $C_{B}>0$ depending on $\alpha$ such that the posterior probability that $\theta \leqslant C_{B} \hat{\theta}_{B}$ is equal to $1-\alpha$.

  [The density of the $\Gamma(k, \lambda)$ distribution is $f(x ; k, \lambda)=\lambda^{k} x^{k-1} e^{-\lambda x} / \Gamma(k)$, for $\left.x>0 .\right]$

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• Paper 4, Section II, E

  #Statistics

  2010

  Consider a collection $X_{1}, \ldots, X_{n}$ of independent random variables with common density function $f(x ; \theta)$ depending on a real parameter $\theta$. What does it mean to say $T$ is a sufficient statistic for $\theta$ ? Prove that if the joint density of $X_{1}, \ldots, X_{n}$ satisfies the factorisation criterion for a statistic $T$, then $T$ is sufficient for $\theta$.

  Let each $X_{i}$ be uniformly distributed on $[-\sqrt{\theta}, \sqrt{\theta}]$. Find a two-dimensional sufficient statistic $T=\left(T_{1}, T_{2}\right)$. Using the fact that $\hat{\theta}=3 X_{1}^{2}$ is an unbiased estimator of $\theta$, or otherwise, find an unbiased estimator of $\theta$ which is a function of $T$ and has smaller variance than $\hat{\theta}$. Clearly state any results you use.

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• Paper 1, Section I, D

  #Variational Principles

  2010

  (a) Define what it means for a function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ to be convex and strictly convex.

  (b) State a necessary and sufficient first-order condition for strict convexity of $f \in C^{1}\left(\mathbb{R}^{n}\right)$, and give, with proof, an example of a function which is strictly convex but with second derivative which is not everywhere strictly positive.

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• Paper 1, Section II, G

  #Algebraic Geometry

  2010

  (i) Let $X=\left\{(x, y) \in \mathbb{C}^{2} \mid x^{2}=y^{3}\right\}$. Show that $X$ is birational to $\mathbf{A}^{1}$, but not isomorphic to it.

  (ii) Let $X$ be an affine variety. Define the dimension of $X$ in terms of the tangent spaces of $X$.

  (iii) Let $f \in k\left[x_{1}, \ldots, x_{n}\right]$ be an irreducible polynomial, where $k$ is an algebraically closed field of arbitrary characteristic. Show that $\operatorname{dim} Z(f)=n-1$.

  [You may assume the Nullstellensatz.]

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• Paper 2, Section II, G

  #Algebraic Geometry

  2010

  Let $X=X_{n, m, r}$ be the set of $n \times m$ matrices of rank at most $r$ over a field $k$. Show that $X_{n, m, r}$ is naturally an affine subvariety of $\mathbf{A}^{n m}$ and that $X_{n, m, r}$ is a Zariski closed subvariety of $X_{n, m, r+1}$.

  Show that if $r<\min (n, m)$, then 0 is a singular point of $X$.

  Determine the dimension of $X_{5,2,1}$.

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• Paper 3, Section II, G

  #Algebraic Geometry

  2010

  (i) Let $X$ be a curve, and $p \in X$ be a smooth point on $X$. Define what a local parameter at $p$ is.

  Now let $f: X \rightarrow Y$ be a rational map to a quasi-projective variety $Y$. Show that if $Y$ is projective, $f$ extends to a morphism defined at $p$.

  Give an example where this fails if $Y$ is not projective, and an example of a morphism $f: \mathbb{C}^{2} \backslash\{0\} \rightarrow \mathbf{P}^{1}$ which does not extend to $0 .$

  (ii) Let $V=Z\left(X_{0}^{8}+X_{1}^{8}+X_{2}^{8}\right)$ and $W=Z\left(X_{0}^{4}+X_{1}^{4}+X_{2}^{4}\right)$ be curves in $\mathbf{P}^{2}$ over a field of characteristic not equal to 2 . Let $\phi: V \rightarrow W$ be the map $\left[X_{0}: X_{1}: X_{2}\right] \mapsto\left[X_{0}^{2}: X_{1}^{2}: X_{2}^{2}\right]$. Determine the degree of $\phi$, and the ramification $e_{p}$ for all $p \in V$.

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• Paper 4, Section II, G

  #Algebraic Geometry

  2010

  Let $E \subseteq \mathbf{P}^{2}$ be the projective curve obtained from the affine curve $y^{2}=\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right)\left(x-\lambda_{3}\right)$, where the $\lambda_{i}$ are distinct and $\lambda_{1} \lambda_{2} \lambda_{3} \neq 0$.

  (i) Show there is a unique point at infinity, $P_{\infty}$.

  (ii) Compute $\operatorname{div}(x), \operatorname{div}(y)$.

  (iii) Show $\mathcal{L}\left(P_{\infty}\right)=k$.

  (iv) Compute $l\left(n P_{\infty}\right)$ for all $n$.

  [You may not use the Riemann-Roch theorem.]

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• Paper 1, Section II, H

  #Algebraic Topology

  2010

  State the path lifting and homotopy lifting lemmas for covering maps. Suppose that $X$ is path connected and locally path connected, that $p_{1}: Y_{1} \rightarrow X$ and $p_{2}: Y_{2} \rightarrow X$ are covering maps, and that $Y_{1}$ and $Y_{2}$ are simply connected. Using the lemmas you have stated, but without assuming the correspondence between covering spaces and subgroups of $\pi_{1}$, prove that $Y_{1}$ is homeomorphic to $Y_{2}$.

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• Paper 2, Section II, $\mathbf{2 1 H}$

  #Algebraic Topology

  2010

  Let $G$ be the finitely presented group $G=\left\langle a, b \mid a^{2} b^{3} a^{3} b^{2}=1\right\rangle$. Construct a path connected space $X$ with $\pi_{1}(X, x) \cong G$. Show that $X$ has a unique connected double cover $\pi: Y \rightarrow X$, and give a presentation for $\pi_{1}(Y, y)$.

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• Paper 3, Section II, H

  #Algebraic Topology

  2010

  Suppose $X$ is a finite simplicial complex and that $H_{*}(X)$ is a free abelian group for each value of $*$. Using the Mayer-Vietoris sequence or otherwise, compute $H_{*}\left(S^{1} \times X\right)$ in terms of $H_{*}(X)$. Use your result to compute $H_{*}\left(T^{n}\right)$.

  [Note that $T^{n}=S^{1} \times \ldots \times S^{1}$, where there are $n$ factors in the product.]

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• Paper 3, Section II, B

  #Applications of Quantum Mechanics

  2010

  State Bloch's theorem for a one dimensional lattice which is invariant under translations by $a$.

  A simple model of a crystal consists of a one-dimensional linear array of identical sites with separation $a$. At the $n$th site the Hamiltonian, neglecting all other sites, is $H_{n}$ and an electron may occupy either of two states, $\phi_{n}(x)$ and $\chi_{n}(x)$, where

  $$H_{n} \phi_{n}(x)=E_{0} \phi_{n}(x), \quad H_{n} \chi_{n}(x)=E_{1} \chi_{n}(x),$$

  and $\phi_{n}$ and $\chi_{n}$ are orthonormal. How are $\phi_{n}(x)$ and $\chi_{n}(x)$ related to $\phi_{0}(x)$ and $\chi_{0}(x)$ ?

  The full Hamiltonian is $H$ and is invariant under translations by $a$. Write trial wavefunctions $\psi(x)$ for the eigenstates of this model appropriate to a tight binding approximation if the electron has probability amplitudes $b_{n}$ and $c_{n}$ to be in the states $\phi_{n}$ and $\chi_{n}$ respectively.

  Assume that the only non-zero matrix elements in this model are, for all $n$,

  $$\begin{aligned} &\left(\phi_{n}, H_{n} \phi_{n}\right)=E_{0}, \quad\left(\chi_{n}, H_{n} \chi_{n}\right)=E_{1} \\ &\left(\phi_{n}, V \phi_{n \pm 1}\right)=\left(\chi_{n}, V \chi_{n \pm 1}\right)=\left(\phi_{n}, V \chi_{n \pm 1}\right)=\left(\chi_{n}, V \phi_{n \pm 1}\right)=-A, \end{aligned}$$

  where $H=H_{n}+V$ and $A>0$. Show that the time-dependent Schrödinger equation governing the amplitudes becomes

  $$\begin{aligned} i \hbar \dot{b}_{n} &=E_{0} b_{n}-A\left(b_{n+1}+b_{n-1}+c_{n+1}+c_{n-1}\right) \\ i \hbar \dot{c}_{n} &=E_{1} c_{n}-A\left(c_{n+1}+c_{n-1}+b_{n+1}+b_{n-1}\right) \end{aligned}$$

  By examining solutions of the form

  $$\left(\begin{array}{l} b_{n} \\ c_{n} \end{array}\right)=\left(\begin{array}{l} B \\ C \end{array}\right) e^{i(k n a-E t / \hbar)}$$

  show that the allowed energies of the electron are two bands given by

  $$E=\frac{1}{2}\left(E_{0}+E_{1}-4 A \cos k a\right) \pm \frac{1}{2} \sqrt{\left(E_{0}-E_{1}\right)^{2}+16 A^{2} \cos ^{2} k a}$$

  Define the Brillouin zone for this system and find the energies at the top and bottom of both bands. Hence, show that the energy gap between the bands is

  $$\Delta E=-4 A+\sqrt{\left(E_{1}-E_{0}\right)^{2}+16 A^{2}}$$

  Show that the wavefunctions $\psi(x)$ satisfy Bloch's theorem.

  Describe briefly what are the crucial differences between insulators, conductors and semiconductors.

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• Paper 4, Section II, $\mathbf{2 1 H}$

  #Algebraic Topology

  2010

  State the Snake Lemma. Explain how to define the boundary map which appears in it, and check that it is well-defined. Derive the Mayer-Vietoris sequence from the Snake Lemma.

  Given a chain complex $C$, let $A \subset C$ be the span of all elements in $C$ with grading greater than or equal to $n$, and let $B \subset C$ be the span of all elements in $C$ with grading less than $n$. Give a short exact sequence of chain complexes relating $A, B$, and $C$. What is the boundary map in the corresponding long exact sequence?

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• Paper 1, Section II, B

  #Applications of Quantum Mechanics

  2010

  Give an account of the variational principle for establishing an upper bound on the ground-state energy, $E_{0}$, of a particle moving in a potential $V(x)$ in one dimension.

  Explain how an upper bound on the energy of the first excited state can be found in the case that $V(x)$ is a symmetric function.

  A particle of mass $2 m=\hbar^{2}$ moves in the potential

  $$V(x)=-V_{0} e^{-x^{2}}, \quad V_{0}>0$$

  Use the trial wavefunction

  $$\psi(x)=e^{-\frac{1}{2} a x^{2}}$$

  where $a$ is a positive real parameter, to establish the upper bound $E_{0} \leqslant E(a)$ for the energy of the ground state, where

  $$E(a)=\frac{1}{2} a-V_{0} \frac{\sqrt{a}}{\sqrt{1+a}} .$$

  Show that, for $a>0, E(a)$ has one zero and find its position.

  Show that the turning points of $E(a)$ are given by

  $$(1+a)^{3}=\frac{V_{0}^{2}}{a}$$

  and deduce that there is one turning point in $a>0$ for all $V_{0}>0$.

  Sketch $E(a)$ for $a>0$ and hence deduce that $V(x)$ has at least one bound state for all $V_{0}>0$.

  For $0<V_{0} \ll 1$ show that

  $$-V_{0}<E_{0} \leqslant \epsilon\left(V_{0}\right),$$

  where $\epsilon\left(V_{0}\right)=-\frac{1}{2} V_{0}^{2}+\mathrm{O}\left(V_{0}^{4}\right)$.

  [You may use the result that $\int_{-\infty}^{\infty} e^{-b x^{2}} d x=\sqrt{\frac{\pi}{b}}$ for $b>0 .$ ]

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• Paper 2, Section II, B

  #Applications of Quantum Mechanics

  2010

  A beam of particles of mass $m$ and momentum $p=\hbar k$ is incident along the $z$-axis. Write down the asymptotic form of the wave function which describes scattering under the influence of a spherically symmetric potential $V(r)$ and which defines the scattering amplitude $f(\theta)$.

  Given that, for large $r$,

  $$e^{i k r \cos \theta} \sim \frac{1}{2 i k r} \sum_{l=0}^{\infty}(2 l+1)\left(e^{i k r}-(-1)^{l} e^{-i k r}\right) P_{l}(\cos \theta),$$

  show how to derive the partial-wave expansion of the scattering amplitude in the form

  $$f(\theta)=\frac{1}{k} \sum_{l=0}^{\infty}(2 l+1) e^{i \delta_{l}} \sin \delta_{l} P_{l}(\cos \theta)$$

  Obtain an expression for the total cross-section, $\sigma$.

  Let $V(r)$ have the form

  $$V(r)=\left\{\begin{array}{cl} -V_{0}, & r<a \\ 0, & r>a \end{array}\right.$$

  where $V_{0}=\frac{\hbar^{2}}{2 m} \gamma^{2}$

  Show that the $l=0$ phase-shift $\delta_{0}$ satisfies

  $$\frac{\tan \left(k a+\delta_{0}\right)}{k a}=\frac{\tan \kappa a}{\kappa a},$$

  where $\kappa^{2}=k^{2}+\gamma^{2}$.

  Assume $\gamma$ to be large compared with $k$ so that $\kappa$ may be approximated by $\gamma$. Show, using graphical methods or otherwise, that there are values for $k$ for which $\delta_{0}(k)=n \pi$ for some integer $n$, which should not be calculated. Show that the smallest value, $k_{0}$, of $k$ for which this condition holds certainly satisfies $k_{0}<3 \pi / 2 a$.

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• Paper 4, Section II, B

  #Applications of Quantum Mechanics

  2010

  The scattering amplitude for electrons of momentum $\hbar \mathbf{k}$ incident on an atom located at the origin is $f(\hat{\mathbf{r}})$ where $\hat{\mathbf{r}}=\mathbf{r} / r$. Explain why, if the atom is displaced by a position vector a, the asymptotic form of the scattering wave function becomes

  $$\psi_{\mathbf{k}}(\mathbf{r}) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+e^{i \mathbf{k} \cdot \mathbf{a}} \frac{e^{i k r^{\prime}}}{r^{\prime}} f\left(\hat{\mathbf{r}}^{\prime}\right) \sim e^{i \mathbf{k} \cdot \mathbf{r}}+e^{i\left(\mathbf{k}-\mathbf{k}^{\prime}\right) \cdot \mathbf{a}} \frac{e^{i k r}}{r} f(\hat{\mathbf{r}}),$$

  where $\mathbf{r}^{\prime}=\mathbf{r}-\mathbf{a}, r^{\prime}=\left|\mathbf{r}^{\prime}\right|, \hat{\mathbf{r}}^{\prime}=\mathbf{r}^{\prime} / r^{\prime}$ and $k=|\mathbf{k}|, \mathbf{k}^{\prime}=k \hat{\mathbf{r}}$. For electrons incident on $N$ atoms in a regular Bravais crystal lattice show that the differential cross-section for scattering in the direction $\hat{\mathbf{r}}$ is

  $$\frac{d \sigma}{d \Omega}=N|f(\hat{\mathbf{r}})|^{2} \Delta\left(\mathbf{k}-\mathbf{k}^{\prime}\right) .$$

  Derive an explicit form for $\Delta(\mathbf{Q})$ and show that it is strongly peaked when $\mathbf{Q} \approx \mathbf{b}$ for $\mathbf{b}$ a reciprocal lattice vector.

  State the Born approximation for $f(\hat{\mathbf{r}})$ when the scattering is due to a potential $V(\mathbf{r})$. Calculate the Born approximation for the case $V(\mathbf{r})=-a \delta(\mathbf{r}) .$

  Electrons with de Broglie wavelength $\lambda$ are incident on a target composed of many randomly oriented small crystals. They are found to be scattered strongly through an angle of $60^{\circ}$. What is the likely distance between planes of atoms in the crystal responsible for the scattering?

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• Paper 1, Section II, I

  #Applied Probability

  2010

  (a) Define what it means to say that $\pi$ is an equilibrium distribution for a Markov chain on a countable state space with Q-matrix $Q=\left(q_{i j}\right)$, and give an equation which is satisfied by any equilibrium distribution. Comment on the possible non-uniqueness of equilibrium distributions.

  (b) State a theorem on convergence to an equilibrium distribution for a continuoustime Markov chain.

  A continuous-time Markov chain $\left(X_{t}, t \geqslant 0\right)$ has three states $1,2,3$ and the Qmatrix $Q=\left(q_{i j}\right)$ is of the form

  $$Q=\left(\begin{array}{ccc} -\lambda_{1} & \lambda_{1} / 2 & \lambda_{1} / 2 \\ \lambda_{2} / 2 & -\lambda_{2} & \lambda_{2} / 2 \\ \lambda_{3} / 2 & \lambda_{3} / 2 & -\lambda_{3} \end{array}\right)$$

  where the rates $\lambda_{1}, \lambda_{2}, \lambda_{3} \in[0, \infty)$ are not all zero.

  [Note that some of the $\lambda_{i}$ may be zero, and those cases may need special treatment.]

  (c) Find the equilibrium distributions of the Markov chain in question. Specify the cases of uniqueness and non-uniqueness.

  (d) Find the limit of the transition matrix $P(t)=\exp (t Q)$ when $t \rightarrow \infty$.

  (e) Describe the jump chain $\left(Y_{n}\right)$ and its equilibrium distributions. If $\widehat{P}$ is the jump probability matrix, find the limit of $\widehat{P}^{n}$ as $n \rightarrow \infty$.

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• Paper 2, Section II, I

  #Applied Probability

  2010

  (a) Let $S_{k}$ be the sum of $k$ independent exponential random variables of rate $k \mu$. Compute the moment generating function $\phi_{S_{k}}(\theta)=\mathbb{E} e^{\theta S_{k}}$ of $S_{k}$. Show that, as $k \rightarrow \infty$, functions $\phi_{S_{k}}(\theta)$ converge to a limit. Describe the random variable $S$ for which the limiting function $\lim _{k \rightarrow \infty} \phi_{S_{k}}(\theta)$ coincides with $\mathbb{E} e^{\theta S}$.

  (b) Define the $M / G / 1$ queue with infinite capacity (sometimes written $M / G / 1 / \infty$ ). Introduce the embedded discrete-time Markov chain $\left(X_{n}\right)$ and write down the recursive relation between $X_{n}$ and $X_{n-1}$.

  Consider, for each fixed $k$ and for $0<\lambda<\mu$, an $\mathrm{M} / \mathrm{G} / 1 / \infty$ queue with arrival rate $\lambda$ and with service times distributed as $S_{k}$. Assume that the queue is empty at time 0 . Let $T_{k}$ be the earliest time at which a customer departs leaving the queue empty. Let $A$ be the first arrival time and $B_{k}=T_{k}-A$ the length of the busy period.

  (c) Prove that the moment generating functions $\phi_{B_{k}}(\theta)=\mathbb{E} e^{\theta B_{k}}$ and $\phi_{S_{k}}(\theta)$ are related by the equation

  $$\phi_{B_{k}}(\theta)=\phi_{S_{k}}\left(\theta-\lambda\left(1-\phi_{B_{k}}(\theta)\right)\right)$$

  (d) Prove that the moment generating functions $\phi_{T_{k}}(\theta)=\mathbb{E} e^{\theta T_{k}}$ and $\phi_{S_{k}}(\theta)$ are related by the equation

  $$\frac{\lambda-\theta}{\lambda} \phi_{T_{k}}(\theta)=\phi_{S_{k}}\left((\lambda-\theta)\left(\phi_{T_{k}}(\theta)-1\right)\right)$$

  (e) Assume that, for all $\theta<\lambda$,

  $$\lim _{k \rightarrow \infty} \phi_{B_{k}}(\theta)=\mathbb{E} e^{\theta B}, \quad \lim _{k \rightarrow \infty} \phi_{T_{k}}(\theta)=\mathbb{E} e^{\theta T},$$

  for some random variables $B$ and $T$. Calculate $\mathbb{E} B$ and $\mathbb{E} T$. What service time distribution do these values correspond to?

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• Paper 3, Section II, I

  #Applied Probability

  2010

  Cars looking for a parking space are directed to one of three unlimited parking lots A, B and C. First, immediately after the entrance, the road forks: one direction is to lot A, the other to B and C. Shortly afterwards, the latter forks again, between B and C. See the diagram below.

  

  The policeman at the first road fork directs an entering car with probability $1 / 3$ to A and with probability $2 / 3$ to the second fork. The policeman at the second fork sends the passing cars to $\mathrm{B}$ or $\mathrm{C}$ alternately: cars $1,3,5, \ldots$ approaching the second fork go to $\mathrm{B}$ and cars $2,4,6, \ldots$ to $\mathrm{C}$.

  Assuming that the total arrival process $(N(t))$ of cars is Poisson of rate $\lambda$, consider the processes $\left(X^{\mathrm{A}}(t)\right),\left(X^{\mathrm{B}}(t)\right)$ and $\left(X^{\mathrm{C}}(t)\right), t \geqslant 0$, where $X^{i}(t)$ is the number of cars directed to lot $i$ by time $t$, for $i=\mathrm{A}, \mathrm{B}, \mathrm{C}$. The times for a car to travel from the first to the second fork, or from a fork to the parking lot, are all negligible.

  (a) Characterise each of the processes $\left(X^{\mathrm{A}}(t)\right),\left(X^{\mathrm{B}}(t)\right)$ and $\left(X^{\mathrm{C}}(t)\right)$, by specifying if it is (i) Poisson, (ii) renewal or (iii) delayed renewal. Correspondingly, specify the rate, the holding-time distribution and the distribution of the delay.

  (b) In the case of a renewal process, determine the equilibrium delay distribution.

  (c) Given $s, t>0$, write down explicit expressions for the probability $\mathbb{P}\left(X^{i}(s)=X^{i}(s+t)\right)$ that the interval $(s, t+s)$ is free of points in the corresponding process, $i=\mathrm{A}, \mathrm{B}, \mathrm{C}$.

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• Paper 4, Section II, I

  #Applied Probability

  2010

  (a) Let $\left(X_{t}\right)$ be an irreducible continuous-time Markov chain on a finite or countable state space. What does it mean to say that the chain is (i) transient, (ii) recurrent, (iii) positive recurrent, (iv) null recurrent? What is the relation between equilibrium distributions and properties (iii) and (iv)?

  A population of microorganisms develops in continuous time; the size of the population is a Markov chain $\left(X_{t}\right)$ with states $0,1,2, \ldots$ Suppose $X_{t}=n$. It is known that after a short time $s$, the probability that $X_{t}$ increased by one is $\lambda(n+1) s+o(s)$ and (if $n \geqslant 1$ ) the probability that the population was exterminated between times $t$ and $t+s$ and never revived by time $t+s$ is $\mu s+o(s)$. Here $\lambda$ and $\mu$ are given positive constants. All other changes in the value of $X_{t}$ have a combined probability $o(s)$.

  (b) Write down the Q-matrix of Markov chain $\left(X_{t}\right)$ and determine if $\left(X_{t}\right)$ is irreducible. Show that $\left(X_{t}\right)$ is non-explosive. Determine the jump chain.

  (c) Now assume that

  $$\mu=\lambda$$

  Determine whether the chain is transient or recurrent, and in the latter case whether it is positive or null recurrent. Answer the same questions for the jump chain. Justify your answers.

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• Paper 1, Section II, C

  #Asymptotic Methods

  2010

  For $\lambda>0$ let

  $$I(\lambda)=\int_{0}^{b} f(x) \mathrm{e}^{-\lambda x} d x, \quad \text { with } \quad 0<b<\infty$$

  Assume that the function $f(x)$ is continuous on $0<x \leqslant b$, and that

  $$f(x) \sim x^{\alpha} \sum_{n=0}^{\infty} a_{n} x^{n \beta}$$

  as $x \rightarrow 0_{+}$, where $\alpha>-1$ and $\beta>0$.

  (a) Explain briefly why in this case straightforward partial integrations in general cannot be applied for determining the asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$.

  (b) Derive with proof an asymptotic expansion for $I(\lambda)$ as $\lambda \rightarrow \infty$.

  (c) For the function

  $$B(s, t)=\int_{0}^{1} u^{s-1}(1-u)^{t-1} d u, \quad s, t>0$$

  obtain, using the substitution $u=e^{-x}$, the first two terms in an asymptotic expansion as $s \rightarrow \infty$. What happens as $t \rightarrow \infty$ ?

  [Hint: The following formula may be useful

  $$\Gamma(y)=\int_{0}^{\infty} x^{y-1} \mathrm{e}^{-x} d t, \quad \text { for } \quad x>0$$

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• Paper 3, Section II, C

  #Asymptotic Methods

  2010

  Consider the ordinary differential equation

  $$y^{\prime \prime}=(|x|-E) y$$

  subject to the boundary conditions $y(\pm \infty)=0$. Write down the general form of the Liouville-Green solutions for this problem for $E>0$ and show that asymptotically the eigenvalues $E_{n}, n \in \mathbb{N}$ and $E_{n}<E_{n+1}$, behave as $E_{n}=\mathrm{O}\left(n^{2 / 3}\right)$ for large $n$.

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• Paper 4, Section II, C

  #Asymptotic Methods

  2010

  (a) Consider for $\lambda>0$ the Laplace type integral

  $$I(\lambda)=\int_{a}^{b} f(t) \mathrm{e}^{-\lambda \phi(t)} d t$$

  for some finite $a, b \in \mathbb{R}$ and smooth, real-valued functions $f(t), \phi(t)$. Assume that the function $\phi(t)$ has a single minimum at $t=c$ with $a<c<b$. Give an account of Laplace's method for finding the leading order asymptotic behaviour of $I(\lambda)$ as $\lambda \rightarrow \infty$ and briefly discuss the difference if instead $c=a$ or $c=b$, i.e. when the minimum is attained at the boundary.

  (b) Determine the leading order asymptotic behaviour of

  $$I(\lambda)=\int_{-2}^{1} \cos t \mathrm{e}^{-\lambda t^{2}} d t$$

  as $\lambda \rightarrow \infty$

  (c) Determine also the leading order asymptotic behaviour when cos $t$ is replaced by $\sin t$ in $(*)$.

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• Paper 1, Section I, D

  #Classical Dynamics

  2010

  A system with coordinates $q_{i}, i=1, \ldots, n$, has the Lagrangian $L\left(q_{i}, \dot{q}_{i}\right)$. Define the energy $E$.

  Consider a charged particle, of mass $m$ and charge $e$, moving with velocity $\mathbf{v}$ in the presence of a magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. The usual vector equation of motion can be derived from the Lagrangian

  $$L=\frac{1}{2} m \mathbf{v}^{2}+e \mathbf{v} \cdot \mathbf{A}$$

  where $\mathbf{A}$ is the vector potential.

  The particle moves in the presence of a field such that

  $$\mathbf{A}=(0, r g(z), 0), \quad g(z)>0$$

  referred to cylindrical polar coordinates $(r, \phi, z)$. Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of $r, \phi$ and $z$.

  Show that, if the particle is projected from the point $\left(r_{0}, \phi_{0}, z_{0}\right)$ with velocity $\left(0,-2(e / m) r_{0} g\left(z_{0}\right), 0\right)$, it will describe a circular orbit provided that $g^{\prime}\left(z_{0}\right)=0$.

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• Paper 2, Section I, D

  #Classical Dynamics

  2010

  Given the form

  $$T=\frac{1}{2} T_{i j} \dot{q}_{i} \dot{q}_{j}, \quad V=\frac{1}{2} V_{i j} q_{i} q_{j}$$

  for the kinetic energy $T$ and potential energy $V$ of a mechanical system, deduce Lagrange's equations of motion.

  A light elastic string of length $4 b$, fixed at both ends, has three particles, each of mass $m$, attached at distances $b, 2 b, 3 b$ from one end. Gravity can be neglected. The particles vibrate with small oscillations transversely to the string, the tension $S$ in the string providing the restoring force. Take the displacements of the particles, $q_{i}, i=1,2,3$, to be the generalized coordinates. Take units such that $m=1, S / b=1$ and show that

  $$V=\frac{1}{2}\left[q_{1}^{2}+\left(q_{1}-q_{2}\right)^{2}+\left(q_{2}-q_{3}\right)^{2}+{q_{3}}^{2}\right]$$

  Find the normal-mode frequencies for this system.

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• Paper 3, Section I, D

  #Classical Dynamics

  2010

  Euler's equations for the angular velocity $\boldsymbol{\omega}=\left(\omega_{1}, \omega_{2}, \omega_{3}\right)$ of a rigid body, viewed in the body frame, are

  $$I_{1} \frac{d \omega_{1}}{d t}=\left(I_{2}-I_{3}\right) \omega_{2} \omega_{3}$$

  and cyclic permutations, where the principal moments of inertia are assumed to obey $I_{1}<I_{2}<I_{3}$.

  Write down two quadratic first integrals of the motion.

  There is a family of solutions $\boldsymbol{\omega}(t)$, unique up to time-translations $t \rightarrow\left(t-t_{0}\right)$, which obey the boundary conditions $\boldsymbol{\omega} \rightarrow(0, \Omega, 0)$ as $t \rightarrow-\infty$ and $\boldsymbol{\omega} \rightarrow(0,-\Omega, 0)$ as $t \rightarrow \infty$, for a given positive constant $\Omega$. Show that, for such a solution, one has

  $$\mathbf{L}^{2}=2 E I_{2},$$

  where $\mathbf{L}$ is the angular momentum and $E$ is the kinetic energy.

  By eliminating $\omega_{1}$ and $\omega_{3}$ in favour of $\omega_{2}$, or otherwise, show that, in this case, the second Euler equation reduces to

  $$\frac{d s}{d \tau}=1-s^{2}$$

  where $s=\omega_{2} / \Omega$ and $\tau=\Omega t\left[\left(I_{1}-I_{2}\right)\left(I_{2}-I_{3}\right) / I_{1} I_{3}\right]^{1 / 2}$. Find the general solution $s(\tau)$.

  [You are not expected to calculate $\omega_{1}(t)$ or $\left.\omega_{3}(t) .\right]$

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• Paper 4, Section I, D

  #Classical Dynamics

  2010

  A system with one degree of freedom has Lagrangian $L(q, \dot{q})$. Define the canonical momentum $p$ and the energy $E$. Show that $E$ is constant along any classical path.

  Consider a classical path $q_{c}(t)$ with the boundary-value data

  $$q_{c}(0)=q_{I}, \quad q_{c}(T)=q_{F}, \quad T>0$$

  Define the action $S_{c}\left(q_{I}, q_{F}, T\right)$ of the path. Show that the total derivative $d S_{c} / d T$ along the classical path obeys

  $$\frac{d S_{c}}{d T}=L$$

  Using Lagrange's equations, or otherwise, deduce that

  $$\frac{\partial S_{c}}{\partial q_{F}}=p_{F}, \quad \frac{\partial S_{c}}{\partial T}=-E,$$

  where $p_{F}$ is the final momentum.

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• Paper 2, Section II, D

  #Classical Dynamics

  2010

  An axially-symmetric top of mass $m$ is free to rotate about a fixed point $O$ on its axis. The principal moments of inertia about $O$ are $A, A, C$, and the centre of gravity $G$ is at a distance $\ell$ from $O$. Define Euler angles $\theta, \phi$ and $\psi$ which specify the orientation of the top, where $\theta$ is the inclination of $O G$ to the upward vertical. Show that there are three conserved quantities for the motion, and give their physical meaning.

  Initially, the top is spinning with angular velocity $n$ about $O G$, with $G$ vertically above $O$, before being disturbed slightly. Show that, in the subsequent motion, $\theta$ will remain close to zero provided $C^{2} n^{2}>4 m g \ell A$, but that if $C^{2} n^{2}<4 m g \ell A$, then $\theta$ will attain a maximum value given by

  $$\cos \theta \simeq\left(C^{2} n^{2} / 2 m g \ell A\right)-1$$

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• Paper 4, Section II, D

  #Classical Dynamics

  2010

  A system is described by the Hamiltonian $H(q, p)$. Define the Poisson bracket $\{f, g\}$ of two functions $f(q, p, t), g(q, p, t)$, and show from Hamilton's equations that

  $$\frac{d f}{d t}=\{f, H\}+\frac{\partial f}{\partial t}$$

  Consider the Hamiltonian

  $$H=\frac{1}{2}\left(p^{2}+\omega^{2} q^{2}\right)$$

  and define

  $$a=(p-i \omega q) /(2 \omega)^{1 / 2}, \quad a^{*}=(p+i \omega q) /(2 \omega)^{1 / 2},$$

  where $i=\sqrt{-1}$. Evaluate $\{a, a\}$ and $\left\{a, a^{*}\right\}$, and show that $\{a, H\}=-i \omega a$ and $\left\{a^{*}, H\right\}=i \omega a^{*}$. Show further that, when $f(q, p, t)$ is regarded as a function of the independent complex variables $a, a^{*}$ and of $t$, one has

  $$\frac{d f}{d t}=i \omega\left(a^{*} \frac{\partial f}{\partial a^{*}}-a \frac{\partial f}{\partial a}\right)+\frac{\partial f}{\partial t}$$

  Deduce that both $\log a^{*}-i \omega t$ and $\log a+i \omega t$ are constant during the motion.

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• Paper 1, Section I, H

  #Coding and Cryptography

  2010

  Explain what is meant by saying that a binary code $\mathcal{C}$ is a decodable code with words $C_{j}$ of length $l_{j}$ for $1 \leqslant j \leqslant n$. Prove the MacMillan inequality which states that, for such a code,

  $$\sum_{j=1}^{n} 2^{-l_{j}} \leqslant 1$$

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• Paper 2, Section I, $4 \mathrm{H}$

  #Coding and Cryptography

  2010

  Describe the standard Hamming code of length 7 , proving that it corrects a single error. Find its weight enumeration polynomial.

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• Paper 3, Section I, $4 \mathrm{H}$

  #Coding and Cryptography

  2010

  What is a linear code? What is a parity check matrix for a linear code? What is the minimum distance $d(C)$ for a linear code $C ?$

  If $C_{1}$ and $C_{2}$ are linear codes having a certain relation (which you should specify), define the bar product $C_{1} \mid C_{2}$. Show that

  $$d\left(C_{1} \mid C_{2}\right)=\min \left\{2 d\left(C_{1}\right), d\left(C_{2}\right)\right\}$$

  If $C_{1}$ has parity check matrix $P_{1}$ and $C_{2}$ has parity check matrix $P_{2}$, find a parity check matrix for $C_{1} \mid C_{2}$.

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• Paper 4, Section I, H

  #Coding and Cryptography

  2010

  What is the discrete logarithm problem?

  Describe the Diffie-Hellman key exchange system for two people. What is the connection with the discrete logarithm problem? Why might one use this scheme rather than just a public key system or a classical (pre-1960) coding system?

  Extend the Diffie-Hellman system to $n$ people using $n(n-1)$ transmitted numbers.

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• Paper 1, Section II, H

  #Coding and Cryptography

  2010

  State and prove Shannon's theorem for the capacity of a noisy memoryless binary symmetric channel, defining the terms you use.

  [You may make use of any form of Stirling's formula and any standard theorems from probability, provided that you state them exactly.]

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• Paper 2, Section II, H

  #Coding and Cryptography

  2010

  The Van der Monde matrix $V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right)$ is the $r \times r$ matrix with $(i, j)$ th entry $x_{i-1}^{j-1}$. Find an expression for $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right)$ as a product. Explain why this expression holds if we work modulo $p$ a prime.

  Show that $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{r-1}\right) \equiv 0$ modulo $p$ if $r>p$, and that there exist $x_{0}, \ldots, x_{p-1}$ such that $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{p-1}\right) \not \equiv 0$. By using Wilson's theorem, or otherwise, find the possible values of $\operatorname{det} V\left(x_{0}, x_{1}, \ldots, x_{p-1}\right)$ modulo $p$.

  The Dark Lord Y'Trinti has acquired the services of the dwarf Trigon who can engrave pairs of very large integers on very small rings. The Dark Lord wishes Trigon to engrave $n$ rings in such a way that anyone who acquires $r$ of the rings and knows the Prime Perilous $p$ can deduce the Integer $N$ of Power, but owning $r-1$ rings will give no information whatsoever. The integers $N$ and $p$ are very large and $p>N$. Advise the Dark Lord.

  For reasons to be explained in the prequel, Trigon engraves an $(n+1)$ st ring with random integers. A band of heroes (who know the Prime Perilous and all the information contained in this question) set out to recover the rings. What, if anything, can they say, with very high probability, about the Integer of Power if they have $r$ rings (possibly including the fake)? What can they say if they have $r+1$ rings? What if they have $r+2$ rings?

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• Paper 1, Section I, D

  #Cosmology

  2010

  What is meant by the expression 'Hubble time'?

  For $a(t)$ the scale factor of the universe and assuming $a(0)=0$ and $a\left(t_{0}\right)=1$, where $t_{0}$ is the time now, obtain a formula for the size of the particle horizon $R_{0}$ of the universe.

  Taking

  $$a(t)=\left(t / t_{0}\right)^{\alpha},$$

  show that $R_{0}$ is finite for certain values of $\alpha$. What might be the physically relevant values of $\alpha$ ? Show that the age of the universe is less than the Hubble time for these values of $\alpha$.

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• Paper 2, Section I, D

  #Cosmology

  2010

  The number density $n=N / V$ for a photon gas in equilibrium is given by

  $$n=\frac{8 \pi}{c^{3}} \int_{0}^{\infty} \frac{\nu^{2}}{e^{h \nu / k T}-1} d \nu$$

  where $\nu$ is the photon frequency. By letting $x=h \nu / k T$, show that

  $$n=\alpha T^{3}$$

  where $\alpha$ is a constant which need not be evaluated.

  The photon entropy density is given by

  $$s=\beta T^{3}$$

  where $\beta$ is a constant. By considering the entropy, explain why a photon gas cools as the universe expands.

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• Paper 3, Section I, D

  #Cosmology

  2010

  Consider a homogenous and isotropic universe with mass density $\rho(t)$, pressure $P(t)$ and scale factor $a(t)$. As the universe expands its energy changes according to the relation $d E=-P d V$. Use this to derive the fluid equation

  $$\dot{\rho}=-3 \frac{\dot{a}}{a}\left(\rho+\frac{P}{c^{2}}\right)$$

  Use conservation of energy applied to a test particle at the boundary of a spherical fluid element to derive the Friedmann equation

  $$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8 \pi}{3} G \rho-\frac{k}{a^{2}} c^{2}$$

  where $k$ is a constant. State any assumption you have made. Briefly state the significance of $k$.

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• Paper 4, Section I, D

  #Cosmology

  2010

  The linearised equation for the growth of density perturbations, $\delta_{\mathbf{k}}$, in an isotropic and homogenous universe is

  $$\ddot{\delta}_{\mathbf{k}}+2 \frac{\dot{a}}{a} \dot{\delta}_{\mathbf{k}}+\left(\frac{c_{s}^{2} \mathbf{k}^{2}}{a^{2}}-4 \pi G \rho\right) \delta_{\mathbf{k}}=0$$

  where $\rho$ is the density of matter, $c_{s}$ the sound speed, $c_{s}^{2}=d P / d \rho$, and $\mathbf{k}$ is the comoving wavevector and $a(t)$ is the scale factor of the universe.

  What is the Jean's length? Discuss its significance for the growth of perturbations.

  Consider a universe filled with pressure-free matter with $a(t)=\left(t / t_{0}\right)^{2 / 3}$. Compute the resulting equation for the growth of density perturbations. Show that your equation has growing and decaying modes and comment briefly on the significance of this fact.

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• Paper 1, Section II, D

  #Cosmology

  2010

  A star has pressure $P(r)$ and mass density $\rho(r)$, where $r$ is the distance from the centre of the star. These quantities are related by the pressure support equation

  $$P^{\prime}=-\frac{G m \rho}{r^{2}}$$

  where $P^{\prime}=d P / d r$ and $m(r)$ is the mass within radius $r$. Use this to derive the virial theorem

  $$E_{\text {grav }}=-3\langle P\rangle V,$$

  where $E_{\text {grav }}$ is the total gravitational potential energy and $\langle P\rangle$ the average pressure.

  The total kinetic energy of a spherically symmetric star is related to $\langle P\rangle$ by

  $$E_{\mathrm{kin}}=\alpha\langle P\rangle V$$

  where $\alpha$ is a constant. Use the virial theorem to determine the condition on $\alpha$ for gravitational binding. By considering the relation between pressure and 'internal energy' $U$ for an ideal gas, determine $\alpha$ for the cases of a) an ideal gas of non-relativistic particles, b) an ideal gas of ultra-relativistic particles.

  Why does your result imply a maximum mass for any star? Briefly explain what is meant by the Chandrasekhar limit.

  A white dwarf is in orbit with a companion star. It slowly accretes matter from the other star until its mass exceeds the Chandrasekhar limit. Briefly explain its subsequent evolution.

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• Paper 3, Section II, D

  #Cosmology

  2010

  The number density for particles in thermal equilibrium, neglecting quantum effects, is

  $$n=g_{s} \frac{4 \pi}{h^{3}} \int p^{2} d p \exp (-(E(p)-\mu) / k T)$$

  where $g_{s}$ is the number of degrees of freedom for the particle with energy $E(p)$ and $\mu$ is its chemical potential. Evaluate $n$ for a non-relativistic particle.

  Thermal equilibrium between two species of non-relativistic particles is maintained by the reaction

  $$a+\alpha \leftrightarrow b+\beta$$

  where $\alpha$ and $\beta$ are massless particles. Evaluate the ratio of number densities $n_{a} / n_{b}$ given that their respective masses are $m_{a}$ and $m_{b}$ and chemical potentials are $\mu_{a}$ and $\mu_{b}$.

  Explain how a reaction like the one above is relevant to the determination of the neutron to proton ratio in the early universe. Why does this ratio not fall rapidly to zero as the universe cools?

  Explain briefly the process of primordial nucleosynthesis by which neutrons are converted into stable helium nuclei. Letting

  $$Y_{H e}=\rho_{H e} / \rho$$

  be the fraction of the universe's helium, compute $Y_{H e}$ as a function of the ratio $r=n_{n} / n_{p}$ at the time of nucleosynthesis.

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• Paper 1, Section II, H

  #Differential Geometry

  2010

  (i) State the definition of smooth manifold with boundary and define the notion of boundary. Show that the boundary $\partial X$ is a manifold (without boundary) with $\operatorname{dim} \partial X=\operatorname{dim} X-1$.

  (ii) Let $0<a<1$ and let $x_{1}, x_{2}, x_{3}, x_{4}$ denote Euclidean coordinates on $\mathbb{R}^{4}$. Show that the set

  $X=\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} \leqslant a\right\} \cap\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\right\} \cap\left\{x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=3 / 2\right\}$

  is a manifold with boundary and compute its dimension. You may appeal to standard results concerning regular values of smooth functions.

  (iii) Determine if the following statements are true or false, giving reasons:

  a. If $X$ and $Y$ are manifolds, $f: X \rightarrow Y$ smooth and $Z \subset Y$ a submanifold of codimension $r$ such that $f$ is not transversal to $Z$, then $f^{-1}(Z)$ is not a submanifold of codimension $r$ in $X$.

  b. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth, then the set of regular values of $f$ is open in $Y$.

  c. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth then the set of critical points is of measure 0 in $X$.

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• Paper 2, Section II, H

  #Differential Geometry

  2010

  (i) State and prove the isoperimetric inequality for plane curves. You may appeal to Wirtinger's inequality as long as you state it precisely.

  (ii) State Fenchel's theorem for curves in space.

  (iii) Let $\alpha: I \rightarrow \mathbb{R}^{2}$ be a closed regular plane curve bounding a region $K$. Suppose $K \supset\left[p_{1}, p_{1}+d_{1}\right] \times\left[p_{2}, p_{2}+d_{2}\right]$, for $d_{1}>0, d_{2}>0$, i.e. $K$ contains a rectangle of dimensions $d_{1}, d_{2}$. Let $k(s)$ denote the signed curvature of $\alpha$ with respect to the inward pointing normal, where $\alpha$ is parametrised anticlockwise. Show that there exists an $s_{0} \in I$ such that $k\left(s_{0}\right) \leqslant \sqrt{\pi /\left(d_{1} d_{2}\right)}$.

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• Paper 3, Section II, H

  #Differential Geometry

  2010

  (i) State and prove the Theorema Egregium.

  (ii) Define the notions principal curvatures, principal directions and umbilical point.

  (iii) Let $S \subset \mathbb{R}^{3}$ be a connected compact regular surface (without boundary), and let $D \subset S$ be a dense subset of $S$ with the following property. For all $p \in D$, there exists an open neighbourhood $\mathcal{U}_{p}$ of $p$ in $S$ such that for all $\theta \in[0,2 \pi), \psi_{p, \theta}\left(\mathcal{U}_{p}\right)=\mathcal{U}_{p}$, where $\psi_{p, \theta}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ denotes rotation by $\theta$ around the line through $p$ perpendicular to $T_{p} S$. Show that $S$ is in fact a sphere.

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• Paper 4, Section II, H

  #Differential Geometry

  2010

  (i) Let $S \subset \mathbb{R}^{3}$ be a regular surface. Define the notions exponential map, geodesic polar coordinates, geodesic circles.

  (ii) State and prove Gauss' lemma.

  (iii) Let $S$ be a regular surface. For fixed $r>0$, and points $p, q$ in $S$, let $S_{r}(p)$, $S_{r}(q)$ denote the geodesic circles around $p, q$, respectively, of radius $r$. Show the following statement: for each $p \in S$, there exists an $r=r(p)>0$ and a neighborhood $\mathcal{U}_{p}$ containing $p$ such that for all $q \in \mathcal{U}_{p}$, the sets $S_{r}(p)$ and $S_{r}(q)$ are smooth 1-dimensional manifolds which intersect transversally. What is the cardinality $\bmod 2$ of $S_{r}(p) \cap S_{r}(q)$ ?

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• Paper 1, Section I, D

  #Dynamical Systems

  2010

  Consider the 2-dimensional flow

  $$\dot{x}=-\mu x+y, \quad \dot{y}=\frac{x^{2}}{1+x^{2}}-\nu y$$

  where $x(t)$ and $y(t)$ are non-negative, the parameters $\mu$ and $\nu$ are strictly positive and $\mu \neq \nu$. Sketch the nullclines in the $x, y$ plane. Deduce that for $\mu<\mu_{c}$ (where $\mu_{c}$ is to be determined) there are three fixed points. Find them and determine their type.

  Sketch the phase portrait for $\mu<\mu_{c}$ and identify, qualitatively on your sketch, the stable and unstable manifolds of the saddle point. What is the final outcome of this system?

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• Paper 2, Section I, D

  #Dynamical Systems

  2010

  Consider the 2-dimensional flow

  $$\dot{x}=\mu\left(\frac{1}{3} x^{3}-x\right)+y, \quad \dot{y}=-x$$

  where the parameter $\mu>0$. Using Lyapunov's approach, discuss the stability of the fixed point and its domain of attraction. Relevant definitions or theorems that you use should be stated carefully, but proofs are not required.

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• Paper 3, Section I, D

  #Dynamical Systems

  2010

  Let $I=[0,1)$. The sawtooth (Bernoulli shift) map $F: I \rightarrow I$ is defined by

  $$F(x)=2 x[\bmod 1]$$

  Describe the effect of $F$ using binary notation. Show that $F$ is continuous on $I$ except at $x=\frac{1}{2}$. Show also that $F$ has $N$-periodic points for all $N \geqslant 2$. Are they stable?

  Explain why $F$ is chaotic, using Glendinning's definition.

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• Paper 4, Section I, D

  #Dynamical Systems

  2010

  Consider the 2-dimensional flow

  $$\dot{x}=y+\frac{1}{4} x\left(1-2 x^{2}-2 y^{2}\right), \quad \dot{y}=-x+\frac{1}{2} y\left(1-x^{2}-y^{2}\right)$$

  Use the Poincaré-Bendixson theorem, which should be stated carefully, to obtain a domain $\mathcal{D}$ in the $x y$-plane, within which there is at least one periodic orbit.

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• Paper 3, Section II, D

  #Dynamical Systems

  2010

  Describe informally the concepts of extended stable manifold theory. Illustrate your discussion by considering the 2-dimensional flow

  $$\dot{x}=\mu x+x y-x^{3}, \quad \dot{y}=-y+y^{2}-x^{2},$$

  where $\mu$ is a parameter with $|\mu| \ll 1$, in a neighbourhood of the origin. Determine the nature of the bifurcation.

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• Paper 4, Section II, D

  #Dynamical Systems

  2010

  Let $I=[0,1]$ and consider continuous maps $F: I \rightarrow I$. Give an informal outline description of the two different bifurcations of fixed points of $F$ that can occur.

  Illustrate your discussion by considering in detail the logistic map

  $$F(x)=\mu x(1-x),$$

  for $\mu \in(0,1+\sqrt{6}]$.

  Describe qualitatively what happens for $\mu \in(1+\sqrt{6}, 4]$.

  [You may assume without proof that

  $$x-F^{2}(x)=x(\mu x-\mu+1)\left(\mu^{2} x^{2}-\mu(\mu+1) x+\mu+1\right)$$

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• Paper 1, Section II, B

  #Electrodynamics

  2010

  The vector potential $A^{\mu}$ is determined by a current density distribution $j^{\mu}$ in the gauge $\partial_{\mu} A^{\mu}=0$ by

  $$\square A^{\mu}=-\mu_{0} j^{\mu}, \quad \square=-\frac{\partial^{2}}{\partial t^{2}}+\nabla^{2},$$

  in units where $c=1$.

  Describe how to justify the result

  $$A^{\mu}(\mathbf{x}, t)=\frac{\mu_{0}}{4 \pi} \int d^{3} x^{\prime} \frac{j^{\mu}\left(\mathbf{x}^{\prime}, t^{\prime}\right)}{\left|\mathbf{x}-\mathbf{x}^{\prime}\right|}, \quad t^{\prime}=t-\left|\mathbf{x}-\mathbf{x}^{\prime}\right|$$

  A plane square loop of thin wire, edge lengths $l$, has its centre at the origin and lies in the $(x, y)$ plane. For $t<0$, no current is flowing in the loop, but at $t=0$ a constant current $I$ is turned on.

  Find the vector potential at the point $(0,0, z)$ as a function of time due to a single edge of the loop.

  What is the electric field due to the entire loop at $(0,0, z)$ as a function of time? Give a careful justification of your answer.

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• Paper 3, Section II, B

  #Electrodynamics

  2010

  A particle of rest-mass $m$, electric charge $q$, is moving relativistically along the path $x^{\mu}(s)$ where $s$ parametrises the path.

  Write down an action for which the extremum determines the particle's equation of motion in an electromagnetic field given by the potential $A^{\mu}(x)$.

  Use your action to derive the particle's equation of motion in a form where $s$ is the proper time.

  Suppose that the electric and magnetic fields are given by

  $$\begin{aligned} \mathbf{E} &=(0,0, E) \\ \mathbf{B} &=(0, B, 0) \end{aligned}$$

  where $E$ and $B$ are constants and $B>E>0$.

  Find $x^{\mu}(s)$ given that the particle starts at rest at the origin when $s=0$.

  Describe qualitatively the motion of the particle.

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• Paper 4, Section II, B

  #Electrodynamics

  2010

  In a superconductor the number density of charge carriers of charge $q$ is $n_{s}$. Suppose that there is a time-independent magnetic field described by the three-vector potential $\mathbf{A} .$

  Derive an expression for the superconducting current.

  Explain how your answer is gauge invariant.

  Suppose that for $z<0$ there is a constant magnetic field $\mathbf{B}_{0}$ in a vacuum and, for $z>0$, there is a uniform superconductor. Derive the magnetic field for $z>0$.

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• Paper 1, Section II, A

  #Fluid Dynamics II

  2010

  Write down the Navier-Stokes equation for the velocity $\mathbf{u}(\mathbf{x}, t)$ of an incompressible viscous fluid of density $\rho$ and kinematic viscosity $\nu$. Cast the equation into dimensionless form. Define rectilinear flow, and explain why the spatial form of any steady rectilinear flow is independent of the Reynolds number.

  (i) Such a fluid is contained between two infinitely long plates at $y=0, y=a$. The lower plate is at rest while the upper plate moves at constant speed $U$ in the $x$ direction. There is an applied pressure gradient $d p / d x=-G \rho \nu$ in the $x$ direction. Determine the flow field.

  (ii) Now there is no applied pressure gradient, but baffles are attached to the lower plate at a distance $L$ from each other $(L \gg a)$, lying between the plates so as to prevent any net volume flux in the $x$ direction. Assuming that far from the baffles the flow is essentially rectilinear, determine the flow field and the pressure gradient in the fluid.

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• Paper 2, Section II, A

  #Fluid Dynamics II

  2010

  What is lubrication theory? Explain the assumptions that go into the theory.

  Viscous fluid with dynamic viscosity $\mu$ and density $\rho$ is contained between two flat plates, which approach each other at uniform speed $V$. The first is fixed at $y=0,-L<x<L$. The second has its ends at $\left(-L, h_{0}-\Delta h-V t\right),\left(L, h_{0}+\Delta h-V t\right)$, where $\Delta h \sim h_{0} \ll L$. There is no flow in the $z$ direction, and all variation in $z$ may be neglected. There is no applied pressure gradient in the $x$ direction.

  Assuming that $V$ is so small that lubrication theory applies, derive an expression for the horizontal volume flux $Q(x)$ at $t=0$, in terms of the pressure gradient. Show that mass conservation implies that $d Q / d x=V$, so that $Q(L)-Q(-L)=2 V L$. Derive another relation between $Q(L)$ and $Q(-L)$ by setting the pressures at $x=\pm L$ to be equal, and hence show that

  $$Q(\pm L)=V L\left(\frac{\Delta h}{h_{0}} \pm 1\right)$$

  Show that lubrication theory applies if $V \ll \mu / h_{0} \rho$.

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• Paper 3, Section II, A

  #Fluid Dynamics II

  2010

  The equation for the vorticity $\omega(x, y)$ in two-dimensional incompressible flow takes the form

  $$\frac{\partial \omega}{\partial t}+u \frac{\partial \omega}{\partial x}+v \frac{\partial \omega}{\partial y}=\nu\left(\frac{\partial^{2} \omega}{\partial x^{2}}+\frac{\partial^{2} \omega}{\partial y^{2}}\right)$$

  where

  $$u=\frac{\partial \psi}{\partial y}, \quad v=-\frac{\partial \psi}{\partial x} \quad \text { and } \quad \omega=-\left(\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{\partial^{2} \psi}{\partial y^{2}}\right)$$

  and $\psi(x, y)$ is the stream function.

  Show that this equation has a time-dependent similarity solution of the form

  $$\psi=C x H(t)^{-1} \phi(\eta), \quad \omega=-C x H(t)^{-3} \phi_{\eta \eta}(\eta) \quad \text { for } \quad \eta=y H(t)^{-1}$$

  if $H(t)=\sqrt{2 C t}$ and $\phi$ satisfies the equation

  $$3 \phi_{\eta \eta}+\eta \phi_{\eta \eta \eta}-\phi_{\eta} \phi_{\eta \eta}+\phi \phi_{\eta \eta \eta}+\frac{1}{R} \phi_{\eta \eta \eta \eta}=0$$

  and $R=C / \nu$ is the effective Reynolds number.

  Show that this solution is appropriate for the problem of two-dimensional flow between the rigid planes $y=\pm H(t)$, and determine the boundary conditions on $\phi$ in that case.

  Verify that $(*)$ has exact solutions, satisfying the boundary conditions, of the form

  $$\phi=\frac{(-1)^{k}}{k \pi} \sin (k \pi \eta)-\eta, \quad k=1,2, \ldots$$

  when $R=k^{2} \pi^{2} / 4$. Sketch this solution when $k$ is large, and discuss whether such solutions are likely to be realised in practice.

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• Paper 4, Section II, A

  #Fluid Dynamics II

  2010

  An axisymmetric incompressible Stokes flow has the Stokes stream function $\Psi(R, \theta)$ in spherical polar coordinates $(R, \theta, \phi)$. Give expressions for the components $u_{R}, u_{\theta}$ of the flow field in terms of $\Psi$. Show that the equation satisfied by $\Psi$ is

  $$\mathcal{D}^{2}\left(\mathcal{D}^{2} \Psi\right)=0, \quad \text { where } \quad \mathcal{D}^{2}=\frac{\partial^{2}}{\partial R^{2}}+\frac{\sin \theta}{R^{2}} \frac{\partial}{\partial \theta}\left(\frac{1}{\sin \theta} \frac{\partial}{\partial \theta}\right)$$

  Fluid is contained between the two spheres $R=a, R=b$, with $b \gg a$. The fluid velocity vanishes on the outer sphere, while on the inner sphere $u_{R}=U \cos \theta, u_{\theta}=0$. It is assumed that Stokes flow applies.

  (i) Show that the Stokes stream function,

  $$\Psi(R, \theta)=a^{2} U \sin ^{2} \theta\left(A\left(\frac{a}{R}\right)+B\left(\frac{R}{a}\right)+C\left(\frac{R}{a}\right)^{2}+D\left(\frac{R}{a}\right)^{4}\right)$$

  is the general solution of $(*)$ proportional to $\sin ^{2} \theta$ and write down the conditions on $A, B, C, D$ that allow all the boundary conditions to be satisfied.

  (ii) Now let $b \rightarrow \infty$, with $|\mathbf{u}| \rightarrow 0$ as $R \rightarrow \infty$. Show that $A=B=1 / 4$ with $C=D=0$.

  (iii) Show that when $b / a$ is very large but finite, then the coefficients have the approximate form

  $$C \approx-\frac{3}{8} \frac{a}{b}, \quad D \approx \frac{1}{8} \frac{a^{3}}{b^{3}}, \quad A \approx \frac{1}{4}-\frac{3}{16} \frac{a}{b}, \quad B \approx \frac{1}{4}+\frac{9}{16} \frac{a}{b}$$

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• Paper 1, Section I, E

  #Further Complex Methods

  2010

  Let the complex-valued function $f(z)$ be analytic in the neighbourhood of the point $z_{0}$ and let $u(x, y)$ be the real part of $f(z)$. Show that

  $$f(z)=2 u\left(\frac{z+\bar{z}_{0}}{2}, \frac{z-\bar{z}_{0}}{2 i}\right)-\overline{f\left(z_{0}\right)}, \quad z=x+i y$$

  Hence find the analytic function whose real part is

  $$e^{-y}[x \cos x-y \sin x]$$

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• Paper 2, Section I, E

  #Further Complex Methods

  2010

  Define

  $$F^{\pm}(x)=\lim _{\epsilon \rightarrow 0} \frac{1}{2 \pi i} \int_{-\infty}^{\infty} \frac{f(t)}{t-(x \pm i \epsilon)} d t, \quad x \in \mathbb{R}$$

  Using the fact that

  $$F^{\pm}(x)=\pm \frac{f(x)}{2}+\frac{1}{2 \pi i} P \int_{-\infty}^{\infty} \frac{f(t)}{t-x} d t, \quad x \in \mathbb{R}$$

  where $P$ denotes the Cauchy principal value, find two complex-valued functions $F^{+}(z)$ and $F^{-}(z)$ which satisfy the following conditions

  1. $F^{+}(z)$ and $F^{-}(z)$ are analytic for $\operatorname{Im} z>0$ and $\operatorname{Im} z<0$ respectively, $z=x+i y$;

  2. $F^{+}(x)-F^{-}(x)=\frac{\sin x}{x}, \quad x \in \mathbb{R}$;

  3. $F^{\pm}(z)=\mathrm{O}\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0$.

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• Paper 3, Section I, E

  #Further Complex Methods

  2010

  Let $\Gamma(z)$ and $\zeta(z)$ denote the gamma and the zeta functions respectively, namely

  $$\begin{aligned} &\Gamma(z)=\int_{0}^{\infty} x^{z-1} e^{-x} d x, \quad \operatorname{Re} z>0 \\ &\zeta(z)=\sum_{m=1}^{\infty} \frac{1}{m^{z}}, \quad \operatorname{Re} z>1 \end{aligned}$$

  By employing a series expansion of $\left(1-e^{-x}\right)^{-2}$, prove the following identity

  $$\int_{0}^{\infty} \frac{x^{z}}{\left(e^{x}-1\right)^{2}} d x=\Gamma(z+1)[\zeta(z)-\zeta(z+1)], \quad \operatorname{Re} z>1$$

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• Paper 4, Section $I$, E

  #Further Complex Methods

  2010

  The hypergeometric function $F(a, b ; c ; z)$ can be expressed in the form

  $$F(a, b ; c ; z)=\frac{\Gamma(c)}{\Gamma(b) \Gamma(c-b)} \int_{0}^{1} t^{b-1}(1-t)^{c-b-1}(1-t z)^{-a} d t$$

  for appropriate restrictions on $c, b, z$.

  Express the following integral in terms of a combination of hypergeometric functions

  $$I(u, A)=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{e^{i t(u+1)}}{e^{i t}+i A} d t, \quad|A|>1$$

  [You may use without proof that $\Gamma(z+1)=z \Gamma(z) .$ ]

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• Paper 1, Section II, E

  #Further Complex Methods

  2010

  Consider the partial differential equation for $u(x, t)$,

  $$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}+\beta \frac{\partial u}{\partial x}, \quad \beta>0, \quad 0<x<\infty, \quad t>0$$

  where $u(x, t)$ is required to vanish rapidly for all $t$ as $x \rightarrow \infty$.

  (i) Verify that the PDE $(*)$ can be written in the following form

  $$\left(e^{-i k x+\left(k^{2}-i \beta k\right) t} u\right)_{t}=\left(e^{-i k x+\left(k^{2}-i \beta k\right) t}\left[(i k+\beta) u+u_{x}\right]\right)_{x}$$

  (ii) Define $\hat{u}(k, t)=\int_{0}^{\infty} e^{-i k x} u(x, t) d x$, which is analytic for $\operatorname{Im} k \leqslant 0$. Determine $\hat{u}(k, t)$ in terms of $\hat{u}(k, 0)$ and also the functions $f_{0}, f_{1}$ defined by

  $$f_{0}(\omega, t)=\int_{0}^{t} e^{-\omega\left(t-t^{\prime}\right)} u\left(0, t^{\prime}\right) d t^{\prime}, \quad f_{1}(\omega, t)=\int_{0}^{t} e^{-\omega\left(t-t^{\prime}\right)} u_{x}\left(0, t^{\prime}\right) d t^{\prime}$$

  (iii) Show that in the inverse transform expression for $u(x, t)$ the integrals involving $f_{0}, f_{1}$ may be transformed to the contour

  $$L=\left\{k \in \mathbb{C}: \operatorname{Re}\left(k^{2}-i \beta k\right)=0, \operatorname{Im} k \geqslant \beta\right\}$$

  By considering $\hat{u}\left(k^{\prime}, t\right)$ where $k^{\prime}=-k+i \beta$ and $k \in L$, show that it is possible to obtain an equation which allows $f_{1}$ to be eliminated.

  (iv) Obtain an integral expression for the solution of $(*)$ subject to the the initialboundary value conditions of given $u(x, 0), u(0, t)$.

  [You need to show that

  $$\int_{L} e^{i k x} \hat{u}\left(k^{\prime}, t\right) d k=0, \quad x>0,$$

  by an appropriate closure of the contour which should be justified.]

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• Paper 2, Section II, 14E

  #Further Complex Methods

  2010

  Let

  $$I(z)=i \oint_{C} \frac{u^{z-1}}{u^{2}-4 u+1} d u$$

  where $C$ is a closed anti-clockwise contour which consists of the unit circle joined to a loop around a branch cut along the negative axis between $-1$ and 0 . Show that

  $$I(z)=F(z)+G(z)$$

  where

  $$F(z)=2 \sin (\pi z) \int_{0}^{1} \frac{x^{z-1}}{x^{2}+4 x+1} d x, \quad \operatorname{Re} z>0$$

  and

  $$G(z)=\frac{1}{2} \int_{-\pi}^{\pi} \frac{e^{i(z-1) \theta}}{1+2 \sin ^{2} \frac{\theta}{2}} d \theta, \quad z \in \mathbb{C}$$

  Evaluate $I(z)$ using Cauchy's theorem. Explain how this may be used to obtain an analytic continuation of $F(z)$ valid for all $z \in \mathbb{C}$.

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• Paper 1, Section II, 18H

  #Galois Theory

  2010

  Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\overline{\mathbb{F}}_{q}$ its algebraic closure.

  (i) Give a non-zero polynomial $P(X)$ in $\mathbb{F}_{q}\left[X_{1}, \ldots, X_{n}\right]$ such that

  $$P\left(\alpha_{1}, \ldots, \alpha_{n}\right)=0 \quad \text { for all } \alpha_{1}, \ldots, \alpha_{n} \in \mathbb{F}_{q}$$

  (ii) Show that every irreducible polynomial $P(X)$ of degree $n>0$ in $\mathbb{F}_{q}[X]$ can be factored in $\overline{\mathbb{F}}_{q}[X]$ as $(X-\alpha)\left(X-\alpha^{q}\right)\left(X-\alpha^{q^{2}}\right) \cdots\left(X-\alpha^{q^{n-1}}\right)$ for some $\alpha \in \overline{\mathbb{F}}_{q}$. What is the splitting field and the Galois group of $P$ over $\mathbb{F}_{q}$ ?

  (iii) Let $n$ be a positive integer and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial. Recall that if $K$ is a field of characteristic prime to $n$, then the set of all roots of $\Phi_{n}$ in $K$ is precisely the set of all primitive $n$-th roots of unity in $K$. Using this fact, prove that if $p$ is a prime number not dividing $n$, then $p$ divides $\Phi_{n}(x)$ in $\mathbb{Z}$ for some $x \in \mathbb{Z}$ if and only if $p=a n+1$ for some integer $a$. Write down $\Phi_{n}$ explicitly for three different values of $n$ larger than 2 , and give an example of $x$ and $p$ as above for each $n$.

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• Paper 2, Section II, H

  #Galois Theory

  2010

  (1) Let $F=\mathbb{Q}(\sqrt[3]{5}, \sqrt{5}, i)$. What is the degree of $F / \mathbb{Q}$ ? Justify your answer.

  (2) Let $F$ be a splitting field of $X^{4}-5$ over $\mathbb{Q}$. Determine the Galois group $\operatorname{Gal}(F / \mathbb{Q})$. Determine all the subextensions of $F / \mathbb{Q}$, expressing each in the form $\mathbb{Q}(x)$ or $\mathbb{Q}(x, y)$ for some $x, y \in F$.

  [Hint: If an automorphism $\rho$ of a field $X$ has order 2 , then for every $x \in X$ the element $x+\rho(x)$ is fixed by $\rho$.]

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• Paper 3, Section II, H

  #Galois Theory

  2010

  Let $K$ be a field of characteristic 0 . It is known that soluble extensions of $K$ are contained in a succession of cyclotomic and Kummer extensions. We will refine this statement.

  Let $n$ be a positive integer. The $n$-th cyclotomic field over a field $K$ is denoted by $K\left(\boldsymbol{\mu}_{n}\right)$. Let $\zeta_{n}$ be a primitive $n$-th root of unity in $K\left(\boldsymbol{\mu}_{n}\right)$.

  (i) Write $\zeta_{3} \in \mathbb{Q}\left(\boldsymbol{\mu}_{3}\right), \zeta_{5} \in \mathbb{Q}\left(\boldsymbol{\mu}_{5}\right)$ in terms of radicals. Write $\mathbb{Q}\left(\boldsymbol{\mu}_{3}\right) / \mathbb{Q}$ and $\mathbb{Q}\left(\boldsymbol{\mu}_{5}\right) / \mathbb{Q}$ as a succession of Kummer extensions.

  (ii) Let $n>1$, and $F:=K\left(\zeta_{1}, \zeta_{2}, \ldots, \zeta_{n-1}\right)$. Show that $F\left(\boldsymbol{\mu}_{n}\right) / F$ can be written as a succession of Kummer extensions, using the structure theorem of finite abelian groups (in other words, roots of unity can be written in terms of radicals). Show that every soluble extension of $K$ is contained in a succession of Kummer extensions.

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• Paper 4, Section II, H

  #Galois Theory

  2010

  Let $K$ be a field of characteristic $\neq 2,3$, and assume that $K$ contains a primitive cubic root of unity $\zeta$. Let $P \in K[X]$ be an irreducible cubic polynomial, and let $\alpha, \beta, \gamma$ be its roots in the splitting field $F$ of $P$ over $K$. Recall that the Lagrange resolvent $x$ of $P$ is defined as $x=\alpha+\zeta \beta+\zeta^{2} \gamma$.

  (i) List the possibilities for the group $\operatorname{Gal}(F / K)$, and write out the set $\{\sigma(x) \mid \sigma \in \operatorname{Gal}(F / K)\}$ in each case.

  (ii) Let $y=\alpha+\zeta \gamma+\zeta^{2} \beta$. Explain why $x^{3}, y^{3}$ must be roots of a quadratic polynomial in $K[X]$. Compute this polynomial for $P=X^{3}+b X+c$, and deduce the criterion to identify $\operatorname{Gal}(F / K)$ through the element $-4 b^{3}-27 c^{2}$ of $K$.

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• Paper 1, Section II, B

  #General Relativity

  2010

  Consider a spacetime $\mathcal{M}$ with a metric $g_{a b}\left(x^{c}\right)$ and a corresponding connection $\Gamma_{b c}^{a}$. Write down the differential equation satisfied by a geodesic $x^{a}(\lambda)$, where $\lambda$ is an affine parameter.

  Show how the requirement that

  $$\delta \int g_{a b}\left(x^{c}\right) \frac{d}{d \lambda} x^{a}(\lambda) \frac{d}{d \lambda} x^{b}(\lambda) d \lambda=0$$

  where $\delta$ denotes variation of the path, gives the geodesic equation and determines $\Gamma_{b c}^{a}$.

  Show that the timelike geodesics for the 2 -manifold with line element

  $$d s^{2}=t^{-2}\left(d x^{2}-d t^{2}\right)$$

  are given by

  $$t^{2}=x^{2}+\alpha x+\beta$$

  where $\alpha$ and $\beta$ are constants.

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• Paper 2, Section II, B

  #General Relativity

  2010

  A vector field $k^{a}$ which satisfies

  $$k_{a ; b}+k_{b ; a}=0$$

  is called a Killing vector field. Prove that $k^{a}$ is a Killing vector field if and only if

  $$k^{c} g_{a b, c}+k_{, b}^{c} g_{a c}+k_{, a}^{c} g_{b c}=0$$

  Prove also that if $V^{a}$ satisfies

  $$V_{; b}^{a} V^{b}=0$$

  then

  $$\left(V^{a} k_{a}\right)_{, b} V^{b}=0$$

  for any Killing vector field $k^{a}$.

  In the two-dimensional space-time with coordinates $x^{a}=(u, v)$ and line element

  $$d s^{2}=-d u^{2}+u^{2} d v^{2}$$

  verify that $(0,1), e^{-v}\left(1, u^{-1}\right)$ and $e^{v}\left(-1, u^{-1}\right)$ are Killing vector fields. Show, by using $(*)$ with $V^{a}$ the tangent vector to a geodesic, that geodesics in this space-time are given by

  $$\alpha e^{v}+\beta e^{-v}=2 \gamma u^{-1}$$

  where $\alpha, \beta$ and $\gamma$ are arbitrary real constants.

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• Paper 4, Section II, B

  #General Relativity

  2010

  The Schwarzschild line element is given by

  $$d s^{2}=-F d t^{2}+F^{-1} d r^{2}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)$$

  where $F=1-r_{s} / r$ and $r_{s}$ is the Schwarzschild radius. Obtain the equation of geodesic motion of photons moving in the equatorial plane, $\theta=\pi / 2$, in the form

  $$\left(\frac{d r}{d \tau}\right)^{2}=E^{2}-\frac{h^{2} F}{r^{2}}$$

  where $\tau$ is proper time, and $E$ and $h$ are constants whose physical significance should be indicated briefly.

  Defining $u=1 / r$ show that light rays are determined by

  $$\left(\frac{d u}{d \phi}\right)^{2}=\left(\frac{1}{b}\right)^{2}-u^{2}+r_{s} u^{3}$$

  where $b=h / E$ and $r_{s}$ may be taken to be small. Show that, to zeroth order in $r_{s}$, a light ray is a straight line passing at distance $b$ from the origin. Show that, to first order in $r_{s}$, the light ray is deflected through an angle $2 r_{s} / b$. Comment briefly on some observational evidence for the result.

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• Paper 1, Section I, F

  #Geometry of Group Actions

  2010

  Explain what it means to say that $G$ is a crystallographic group of isometries of the Euclidean plane and that $\bar{G}$ is its point group. Prove the crystallographic restriction: a rotation in such a point group $\bar{G}$ must have order $1,2,3,4$ or 6 .

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• Paper 2, Section I, F

  #Geometry of Group Actions

  2010

  Show that a map $T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is an isometry for the Euclidean metric on the plane $\mathbb{R}^{2}$ if and only if there is a vector $\boldsymbol{v} \in \mathbb{R}^{2}$ and an orthogonal linear map $B \in \mathrm{O}(2)$ with

  $$T(\boldsymbol{x})=B(\boldsymbol{x})+\boldsymbol{v} \quad \text { for all } \boldsymbol{x} \in \mathbb{R}^{2}$$

  When $T$ is an isometry with $\operatorname{det} B=-1$, show that $T$ is either a reflection or a glide reflection.

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• Paper 3, Section I, F

  #Geometry of Group Actions

  2010

  Let $U$ be a "triangular" region in the unit disc $\mathbb{D}$ bounded by three hyperbolic geodesics $\gamma_{1}, \gamma_{2}, \gamma_{3}$ that do not meet in $\mathbb{D}$ nor on its boundary. Let $J_{k}$ be inversion in $\gamma_{k}$ and set

  $$A=J_{2} \circ J_{1} ; \quad B=J_{3} \circ J_{2} .$$

  Let $G$ be the group generated by the Möbius transformations $A$ and $B$. Describe briefly a fundamental set for the group $G$ acting on $\mathbb{D}$.

  Prove that $G$ is a free group on the two generators $A$ and $B$. Describe the quotient surface $\mathbb{D} / G$.

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• Paper 4, Section I, F

  #Geometry of Group Actions

  2010

  Define loxodromic transformations and explain how to determine when a Möbius transformation

  $$T: z \mapsto \frac{a z+b}{c z+d} \quad \text { with } \quad a d-b c=1$$

  is loxodromic.

  Show that any Möbius transformation that maps a disc $\Delta$ onto itself cannot be loxodromic.

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• Paper 1, Section II, F

  #Geometry of Group Actions

  2010

  For which circles $\Gamma$ does inversion in $\Gamma$ interchange 0 and $\infty$ ?

  Let $\Gamma$ be a circle that lies entirely within the unit $\operatorname{disc} \mathbb{D}=\{z \in \mathbb{C}:|z|<1\} .$ Let $K$ be inversion in this circle $\Gamma$, let $J$ be inversion in the unit circle, and let $T$ be the Möbius transformation $K \circ J$. Show that, if $z_{0}$ is a fixed point of $T$, then

  $$J\left(z_{0}\right)=K\left(z_{0}\right)$$

  and this point is another fixed point of $T$.

  By applying a suitable isometry of the hyperbolic plane $\mathbb{D}$, or otherwise, show that $\Gamma$ is the set of points at a fixed hyperbolic distance from some point of $\mathbb{D}$.

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• Paper 4, Section II, F

  #Geometry of Group Actions

  2010

  Explain briefly how Möbius transformations of the Riemann sphere are extended to give isometries of the unit ball $B^{3} \subset \mathbb{R}^{3}$ for the hyperbolic metric.

  Which Möbius transformations have extensions that fix the origin in $B^{3}$ ?

  For which Möbius transformations $T$ can we find a hyperbolic line in $B^{3}$ that $T$ maps onto itself? For which of these Möbius transformations is there only one such hyperbolic line?

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• Paper 1, Section II, F

  #Graph Theory

  2010

  (a) Define the Ramsey number $R(s)$. Show that for all integers $s \geqslant 2$ the Ramsey number $R(s)$ exists and that $R(s) \leqslant 4^{s}$.

  (b) For any graph $G$, let $R(G)$ denote the least positive integer $n$ such that in any red-blue colouring of the edges of the complete graph $K_{n}$ there must be a monochromatic copy of $G$.

  (i) How do we know that $R(G)$ exists for every graph $G$ ?

  (ii) Let $s$ be a positive integer. Show that, whenever the edge of $K_{2 s}$ are red-blue coloured, there must be a monochromatic copy of the complete bipartite graph $K_{1, s}$.

  (iii) Suppose $s$ is odd. By exhibiting a suitable colouring of $K_{2 s-1}$, show that $R\left(K_{1, s}\right)=2 s$.

  (iv) Suppose instead $s$ is even. What is $R\left(K_{1, s}\right) ?$ Justify your answer.

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• Paper 2, Section II, F

  #Graph Theory

  2010

  Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. What does it mean to say that $G$ contains a matching from $X$ to $Y$ ?

  State and prove Hall's Marriage Theorem, giving a necessary and sufficient condition for $G$ to contain a matching from $X$ to $Y$.

  Now assume that $G$ does contain a matching (from $X$ to $Y$ ). For a subset $A \subset X$, $\Gamma(A)$ denotes the set of vertices adjacent to some vertex in $A$.

  (i) Suppose $|\Gamma(A)|>|A|$ for every $A \subset X$ with $A \neq \emptyset, X$. Show that every edge of $G$ is contained in a matching.

  (ii) Suppose that every edge of $G$ is contained in a matching and that $G$ is connected. Show that $|\Gamma(A)|>|A|$ for every $A \subset X$ with $A \neq \emptyset, X$.

  (iii) For each $n \geqslant 2$, give an example of $G$ with $|X|=n$ such that every edge is contained in a matching but $|\Gamma(A)|=|A|$ for some $A \subset X$ with $A \neq \emptyset, X$.

  (iv) Suppose that every edge of $G$ is contained in a matching. Must every pair of independent edges in $G$ be contained in a matching? Give a proof or counterexample as appropriate.

  [No form of Menger's Theorem or of the Max-Flow-Min-Cut Theorem may be assumed without proof.]

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• Paper 3, Section II, F

  #Graph Theory

  2010

  Let $G$ be a graph of order $n$. Show that $G$ must contain an independent set of $\left\lceil\sum_{v \in G} \frac{1}{d(v)+1}\right]$ vertices (where $\lceil x\rceil$ denotes the least integer $\left.\geqslant x\right)$.

  [Hint: take a random ordering of the vertices of $G$, and consider the set of those vertices which are adjacent to no earlier vertex in the ordering.]

  Fix an integer $m<n$ with $m$ dividing $n$, and suppose that $e(G)=m\left(\begin{array}{c}n / m \\ 2\end{array}\right)$.

  (i) Deduce that $G$ must contain an independent set of $m$ vertices.

  (ii) Must $G$ contain an independent set of $m+1$ vertices?

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• Paper 4, Section II, F

  #Graph Theory

  2010

  State Euler's formula relating the number of vertices, edges and faces in a drawing of a connected planar graph. Deduce that every planar graph has chromatic number at most $5 .$

  Show also that any triangle-free planar graph has chromatic number at most 4 .

  Suppose $G$ is a planar graph which is minimal 5 -chromatic; that is to say, $\chi(G)=5$ but if $H$ is a subgraph of $G$ with $H \neq G$ then $\chi(H)<5$. Prove that $\delta(G) \geqslant 5$. Does this remain true if we drop the assumption that $G$ is planar? Justify your answer.

  [The Four Colour Theorem may not be assumed.]

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• Paper 1, Section II, E

  #Integrable Systems

  2010

  Define a Poisson structure on an open set $U \subset \mathbb{R}^{n}$ in terms of an anti-symmetric matrix $\omega^{a b}: U \longrightarrow \mathbb{R}$, where $a, b=1, \cdots, n$. By considering the Poisson brackets of the coordinate functions $x^{a}$ show that

  $$\sum_{d=1}^{n}\left(\omega^{d c} \frac{\partial \omega^{a b}}{\partial x^{d}}+\omega^{d b} \frac{\partial \omega^{c a}}{\partial x^{d}}+\omega^{d a} \frac{\partial \omega^{b c}}{\partial x^{d}}\right)=0$$

  Now set $n=3$ and consider $\omega^{a b}=\sum_{c=1}^{3} \varepsilon^{a b c} x^{c}$, where $\varepsilon^{a b c}$ is the totally antisymmetric symbol on $\mathbb{R}^{3}$ with $\varepsilon^{123}=1$. Find a non-constant function $f: \mathbb{R}^{3} \longrightarrow \mathbb{R}$ such that

  $$\left\{f, x^{a}\right\}=0, \quad a=1,2,3$$

  Consider the Hamiltonian

  $$H\left(x^{1}, x^{2}, x^{3}\right)=\frac{1}{2} \sum_{a, b=1}^{3} M^{a b} x^{a} x^{b}$$

  where $M^{a b}$ is a constant symmetric matrix and show that the Hamilton equations of motion with $\omega^{a b}=\sum_{c=1}^{3} \varepsilon^{a b c} x^{c}$ are of the form

  $$\dot{x}^{a}=\sum_{b, c=1}^{3} Q^{a b c} x^{b} x^{c},$$

  where the constants $Q^{a b c}$ should be determined in terms of $M^{a b}$.

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• Paper 2, Section II, E

  #Integrable Systems

  2010

  Consider the Gelfand-Levitan-Marchenko (GLM) integral equation

  $$K(x, y)+F(x+y)+\int_{x}^{\infty} K(x, z) F(z+y) d z=0$$

  with $F(x)=\sum_{1}^{N} \beta_{n} e^{-c_{n} x}$, where $c_{1}, \ldots, c_{N}$ are positive constants and $\beta_{1}, \ldots, \beta_{N}$ are constants. Consider separable solutions of the form

  $$K(x, y)=\sum_{n=1}^{N} K_{n}(x) e^{-c_{n} y}$$

  and reduce the GLM equation to a linear system

  $$\sum_{m=1}^{N} A_{n m}(x) K_{m}(x)=B_{n}(x)$$

  where the matrix $A_{n m}(x)$ and the vector $B_{n}(x)$ should be determined.

  How is $K$ related to solutions of the $K d V$ equation?

  Set $N=1, c_{1}=c, \beta_{1}=\beta \exp \left(8 c^{3} t\right)$ where $c, \beta$ are constants. Show that the corresponding one soliton solution of the $\mathrm{KdV}$ equation is given by

  $$u(x, t)=-\frac{4 \beta_{1} c e^{-2 c x}}{\left(1+\left(\beta_{1} / 2 c\right) e^{-2 c x}\right)^{2}}$$

  [You may use any facts about the Inverse Scattering Transform without proof.]

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• Paper 3, Section II, E

  #Integrable Systems

  2010

  Consider a vector field

  $$V=\alpha x \frac{\partial}{\partial x}+\beta t \frac{\partial}{\partial t}+\gamma v \frac{\partial}{\partial v}$$

  on $\mathbb{R}^{3}$, where $\alpha, \beta$ and $\gamma$ are constants. Find the one-parameter group of transformations generated by this vector field.

  Find the values of the constants $(\alpha, \beta, \gamma)$ such that $V$ generates a Lie point symmetry of the modified $\mathrm{KdV}$ equation ( $\mathrm{KKdV}$ )

  $$v_{t}-6 v^{2} v_{x}+v_{x x x}=0, \quad \text { where } \quad v=v(x, t)$$

  Show that the function $u=u(x, t)$ given by $u=v^{2}+v_{x}$ satisfies the KdV equation and find a Lie point symmetry of $\mathrm{KdV}$ corresponding to the Lie point symmetry of $\mathrm{mKdV}$ which you have determined from $V$.

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• Paper 1, Section II, H

  #Linear Analysis

  2010

  a) State and prove the Banach-Steinhaus Theorem.

  [You may use the Baire Category Theorem without proving it.]

  b) Let $X$ be a (complex) normed space and $S \subset X$. Prove that if $\{f(x): x \in S\}$ is a bounded set in $\mathbb{C}$ for every linear functional $f \in X^{*}$ then there exists $K \geqslant 0$ such that $\|x\| \leqslant K$ for all $x \in S .$

  [You may use here the following consequence of the Hahn-Banach Theorem without proving it: for a given $x \in X$, there exists $f \in X^{*}$ with $\|f\|=1$ and $|f(x)|=\|x\|$.]

  c) Conclude that if two norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ on a (complex) vector space $V$ are not equivalent, there exists a linear functional $f: V \rightarrow \mathbb{C}$ which is continuous with respect to one of the two norms, and discontinuous with respect to the other.

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• Paper 2, Section II, H

  #Linear Analysis

  2010

  For a sequence $x=\left(x_{1}, x_{2}, \ldots\right)$ with $x_{j} \in \mathbb{C}$ for all $j \geqslant 1$, let

  $$\|x\|_{\infty}:=\sup _{j \geqslant 1}\left|x_{j}\right|$$

  and $\ell^{\infty}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C}\right.$ for all $j \geqslant 1$ and $\left.\|x\|_{\infty}<\infty\right\}$.

  a) Prove that $\ell^{\infty}$ is a Banach space.

  b) Define

  $$c_{0}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: \lim _{j \rightarrow \infty} x_{j}=0\right\}$$

  and

  $$\ell^{1}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \text { for all } j \in \mathbb{N} \text { and }\|x\|_{1}=\sum_{\ell=1}^{\infty}\left|x_{\ell}\right|<\infty\right\}$$

  Show that $c_{0}$ is a closed subspace of $\ell^{\infty}$. Show that $c_{0}^{*} \simeq \ell^{1}$.

  [Hint: find an isometric isomorphism from $\ell^{1}$ to $\left.c_{0}^{*} .\right]$

  c) Let

  $$c_{00}=\left\{x=\left(x_{1}, x_{2}, \ldots\right) \in \ell^{\infty}: x_{j}=0 \text { for all } j \text { large enough }\right\} .$$

  Is $c_{00}$ a closed subspace of $\ell^{\infty} ?$ If not, what is the closure of $c_{00} ?$

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• Paper 3, Section II, H

  #Linear Analysis

  2010

  State and prove the Stone-Weierstrass theorem for real-valued functions.

  [You may use without proof the fact that the function $s \rightarrow|s|$ can be uniformly approximated by polynomials on $[-1,1] .]$

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• Paper 4, Section II, H

  #Linear Analysis

  2010

  Let $X$ be a Banach space.

  a) What does it mean for a bounded linear map $T: X \rightarrow X$ to be compact?

  b) Let $\mathcal{B}(X)$ be the Banach space of all bounded linear maps $S: X \rightarrow X$. Let $\mathcal{B}_{0}(X)$ be the subset of $\mathcal{B}(X)$ consisting of all compact operators. Show that $\mathcal{B}_{0}(X)$ is a closed subspace of $\mathcal{B}(X)$. Show that, if $S \in \mathcal{B}(X)$ and $T \in \mathcal{B}_{0}(X)$, then $S T, T S \in \mathcal{B}_{0}(X)$.

  c) Let

  $$X=\ell^{2}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \quad \text { and } \quad\|x\|_{2}^{2}=\sum_{j=1}^{\infty}\left|x_{j}\right|^{2}<\infty\right\}$$

  and $T: X \rightarrow X$ be defined by

  $$(T x)_{k}=\frac{x_{k+1}}{k+1} .$$

  Is $T$ compact? What is the spectrum of $T ?$ Explain your answers.

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• Paper 1, Section II, G

  #Logic and Set Theory

  2010

  Show that $\aleph_{\alpha}^{2}=\aleph_{\alpha}$ for all $\alpha$.

  An infinite cardinal $m$ is called regular if it cannot be written as a sum of fewer than $m$ cardinals each of which is smaller than $m$. Prove that $\aleph_{0}$ and $\aleph_{1}$ are regular.

  Is $\aleph_{2}$ regular? Is $\aleph_{\omega}$ regular? Justify your answers.

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• Paper 2, Section II, G

  #Logic and Set Theory

  2010

  Let $\alpha$ be a non-zero ordinal. Prove that there exists a greatest ordinal $\beta$ such that $\omega^{\beta} \leqslant \alpha$. Explain why there exists an ordinal $\gamma$ with $\omega^{\beta}+\gamma=\alpha$. Prove that $\gamma$ is unique, and that $\gamma<\alpha$.

  A non-zero ordinal $\alpha$ is called decomposable if it can be written as the sum of two smaller non-zero ordinals. Deduce that if $\alpha$ is not a power of $\omega$ then $\alpha$ is decomposable.

  Conversely, prove that if $\alpha$ is a power of $\omega$ then $\alpha$ is not decomposable.

  [Hint: consider the cases $\alpha=\omega^{\beta}$ ( $\beta$ a successor) and $\alpha=\omega^{\beta}$ ( $\beta$ a limit) separately.]

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• Paper 3, Section II, G

  #Logic and Set Theory

  2010

  Define the sets $V_{\alpha}, \alpha \in O N$. What is meant by the rank of a set?

  Explain briefly why, for every $\alpha$, there exists a set of rank $\alpha$.

  Let $x$ be a transitive set of rank $\alpha$. Show that $x$ has an element of rank $\beta$ for every $\beta<\alpha$.

  For which $\alpha$ does there exist a finite set of rank $\alpha$ ? For which $\alpha$ does there exist a finite transitive set of rank $\alpha$ ? Justify your answers.

  [Standard properties of rank may be assumed.]

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• Paper 4, Section II, G

  #Logic and Set Theory

  2010

  State and prove the Completeness Theorem for Propositional Logic.

  [You do not need to give definitions of the various terms involved. You may assume that the set of primitive propositions is countable. You may also assume the Deduction Theorem.]

  Explain briefly how your proof should be modified if the set of primitive propositions is allowed to be uncountable.

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• Paper 1, Section I, A

  #Mathematical Biology

  2010

  A delay model for a population $N_{t}$ consists of

  $$N_{t+1}=\frac{r N_{t}}{1+b N_{t-1}^{2}}$$

  where $t$ is discrete time, $r>1$ and $b>0$. Investigate the linear stability about the positive steady state $N^{*}$. Show that $r=2$ is a bifurcation value at which the steady state bifurcates to a periodic solution of period 6 .

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• Paper 2, Section $I$, A

  #Mathematical Biology

  2010

  The population of a certain species subjected to a specific kind of predation is modelled by the difference equation

  $$u_{t+1}=a \frac{u_{t}^{2}}{b^{2}+u_{t}^{2}}, \quad a>0$$

  Determine the equilibria and show that if $a^{2}>4 b^{2}$ it is possible for the population to be driven to extinction if it becomes less than a critical size which you should find. Explain your reasoning by means of a cobweb diagram.

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• Paper 3, Section I, A

  #Mathematical Biology

  2010

  A population of aerobic bacteria swims in a laterally-infinite layer of fluid occupying $-\infty<x<\infty,-\infty<y<\infty$, and $-d / 2<z<d / 2$, with the top and bottom surfaces in contact with air. Assuming that there is no fluid motion and that all physical quantities depend only on $z$, the oxygen concentration $c$ and bacterial concentration $n$ obey the coupled equations

  $$\begin{aligned} \frac{\partial c}{\partial t} &=D_{c} \frac{\partial^{2} c}{\partial z^{2}}-k n \\ \frac{\partial n}{\partial t} &=D_{n} \frac{\partial^{2} n}{\partial z^{2}}-\frac{\partial}{\partial z}\left(\mu n \frac{\partial c}{\partial z}\right) \end{aligned}$$

  Consider first the case in which there is no chemotaxis, so $n$ has the spatially-uniform value $\bar{n}$. Find the steady-state oxygen concentration consistent with the boundary conditions $c(\pm d / 2)=c_{0}$. Calculate the Fick's law flux of oxygen into the layer and justify your answer on physical grounds.

  Now allowing chemotaxis and cellular diffusion, show that the equilibrium oxygen concentration satisfies

  $$\frac{d^{2} c}{d z^{2}}-\frac{k n_{0}}{D_{c}} \exp \left(\mu c / D_{n}\right)=0$$

  where $n_{0}$ is a suitable normalisation constant that need not be found.

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• Paper 4, Section I, A

  #Mathematical Biology

  2010

  A concentration $u(x, t)$ obeys the differential equation

  $$\frac{\partial u}{\partial t}=D u_{x x}+f(u)$$

  in the domain $0 \leqslant x \leqslant L$, with boundary conditions $u(0, t)=u(L, t)=0$ and initial condition $u(x, 0)=u_{0}(x)$, and where $D$ is a positive constant. Assume $f(0)=0$ and $f^{\prime}(0)>0$. Linearising the dynamics around $u=0$, and representing $u(x, t)$ as a suitable Fourier expansion, show that the condition for the linear stability of $u=0$ can be expressed as the following condition on the domain length

  $$L<\pi\left[\frac{D}{f^{\prime}(0)}\right]^{1 / 2}$$

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• Paper 2, Section II, A

  #Mathematical Biology

  2010

  The radially symmetric spread of an insect population density $n(r, t)$ in the plane is described by the equation

  $$\frac{\partial n}{\partial t}=\frac{D_{0}}{r} \frac{\partial}{\partial r}\left[r\left(\frac{n}{n_{0}}\right)^{2} \frac{\partial n}{\partial r}\right]$$

  Suppose $Q$ insects are released at $r=0$ at $t=0$. We wish to find a similarity solution to $(*)$ in the form

  $$n(r, t)=\frac{n_{0}}{\lambda^{2}(t)} F\left(\frac{r}{r_{0} \lambda(t)}\right)$$

  Show first that the PDE $(*)$ reduces to an ODE for $F$ if $\lambda(t)$ obeys the equation

  $$\lambda^{5} \frac{d \lambda}{d t}=C \frac{D_{0}}{r_{0}^{2}}$$

  where $C$ is an arbitrary constant (that may be set to unity), and then obtain $\lambda(t)$ and $F$ such that $F(0)=1$ and $F(\xi)=0$ for $\xi \geqslant 1$. Determine $r_{0}$ in terms of $n_{0}$ and $Q$. Sketch the function $n(r, t)$ at various times to indicate its qualitative behaviour.

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• Paper 3, Section II, A

  #Mathematical Biology

  2010

  Consider an epidemic model in which $S(x, t)$ is the local population density of susceptibles and $I(x, t)$ is the density of infectives

  $$\begin{aligned} &\frac{\partial S}{\partial t}=-r I S \\ &\frac{\partial I}{\partial t}=D \frac{\partial^{2} I}{\partial x^{2}}+r I S-a I \end{aligned}$$

  where $r, a$, and $D$ are positive. If $S_{0}$ is a characteristic population value, show that the rescalings $I / S_{0} \rightarrow I, S / S_{0} \rightarrow S,\left(r S_{0} / D\right)^{1 / 2} x \rightarrow x, r S_{0} t \rightarrow t$ reduce this system to

  $$\begin{aligned} &\frac{\partial S}{\partial t}=-I S \\ &\frac{\partial I}{\partial t}=\frac{\partial^{2} I}{\partial x^{2}}+I S-\lambda I \end{aligned}$$

  where $\lambda$ should be found.

  Travelling wavefront solutions are of the form $S(x, t)=S(z), I(x, t)=I(z)$, where $z=x-c t$ and $c$ is the wave speed, and we seek solutions with boundary conditions $S(\infty)=1, S^{\prime}(\infty)=0, I(\infty)=I(-\infty)=0$. Under the travelling-wave assumption reduce the rescaled PDEs to ODEs, and show by linearisation around the leading edge of the advancing front that the requirement that $I$ be non-negative leads to the condition $\lambda<1$ and hence the wave speed relation

  $$c \geqslant 2(1-\lambda)^{1 / 2}, \quad \lambda<1$$

  Using the two ODEs you have obtained, show that the surviving susceptible population fraction $\sigma=S(-\infty)$ after the passage of the front satisfies

  $$\sigma-\lambda \ln \sigma=1,$$

  and sketch $\sigma$ as a function of $\lambda$.

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• Paper 1, Section II, G

  #Number Fields

  2010

  Suppose that $m$ is a square-free positive integer, $m \geqslant 5, m \not \equiv 1 \quad(\bmod 4)$. Show that, if the class number of $K=\mathbb{Q}(\sqrt{-m})$ is prime to 3 , then $x^{3}=y^{2}+m$ has at most two solutions in integers. Assume the $m$ is even.

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• Paper 2, Section II, G

  #Number Fields

  2010

  Calculate the class group of the field $\mathbb{Q}(\sqrt{-14})$.

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• Paper 4, Section II, G

  #Number Fields

  2010

  Suppose that $\alpha$ is a zero of $x^{3}-x+3$ and that $K=\mathbb{Q}(\alpha)$. Show that $[K: \mathbb{Q}]=3$. Show that $O_{K}$, the ring of integers in $K$, is $O_{K}=\mathbb{Z}[\alpha]$.

  [You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of $x^{3}+p x+q$ is $-4 p^{3}-27 q^{2}$.]

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• Paper 1, Section I, G

  #Number Theory

  2010

  (i) Let $N$ be an integer $\geqslant 2$. Define the addition and multiplication on the set of congruence classes modulo $N$.

  (ii) Let an integer $M \geqslant 1$ have expansion to the base 10 given by $a_{s} \ldots a_{0}$. Prove that 11 divides $M$ if and only if $\sum_{i=0}^{s}(-1)^{i} a_{i}$ is divisible by 11 .

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• Paper 2, Section I, G

  #Number Theory

  2010

  Let $p$ be an odd prime number. If $n$ is an integer prime to $p$, define $\left(\frac{n}{p}\right)$.

  (i) Prove that $\chi(n)=\left(\frac{n}{p}\right)$ defines a homomorphism from $(\mathbb{Z} / p \mathbb{Z})^{\times}$to the group $\{\pm 1\}$. What is the value of $\chi(-1) ?$

  (ii) If $p \equiv 1 \bmod 4$, prove that $\sum_{n=1}^{p-1} \chi(n) n=0$.

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• Paper 3, Section I, G

  #Number Theory

  2010

  (i) Let $M$ and $N$ be positive integers, such that $N$ is not a perfect square. If $M<\sqrt{N}$, show that every solution of the equation

  $$x^{2}-N y^{2}=M$$

  in positive integers $x, y$ comes from some convergent of the continued fraction of $\sqrt{N}$.

  (ii) Find a solution in positive integers $x, y$ of

  $$x^{2}-29 y^{2}=5$$

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• Paper 4, Section I, G

  #Number Theory

  2010

  Let $p$ be a prime number, and put

  $$a_{k}=k p, \quad N_{k}=a_{k}^{p}-1 \quad(k=1,2, \ldots)$$

  Prove that $a_{k}$ has exact order $p$ modulo $N_{k}$ for all $k \geqslant 1$, and deduce that $N_{k}$ must be divisible by a prime $q$ with $q \equiv 1 \quad(\bmod p)$. By making a suitable choice of $k$, prove that there are infinitely many primes $q$ with $q \equiv 1 \quad(\bmod p)$.

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• Paper 3, Section II, G

  #Number Theory

  2010

  State precisely the Miller-Rabin primality test.

  (i) Let $p$ be a prime $\geqslant 5$, and define

  $$N=\frac{4^{p}-1}{3}$$

  Prove that $N$ is a composite odd integer, and that $N$ is a pseudo-prime to the base 2 .

  (ii) Let $M$ be an odd integer greater than 1 such that $M$ is a pseudo-prime to the base 2 . Prove that $2^{M}-1$ is always a strong pseudo-prime to the base 2 .

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• Paper 4, Section II, G

  #Number Theory

  2010

  Let $\mathcal{S}$ be the set of all positive definite binary quadratic forms with integer coefficients. Define the action of the group $S L_{2}(\mathbb{Z})$ on $\mathcal{S}$, and prove that equivalent forms under this action have the same discriminant.

  Find necessary and sufficient conditions for an odd positive integer $n$, prime to 35 , to be properly represented by at least one of the two forms

  $$x^{2}+x y+9 y^{2}, \quad 3 x^{2}+x y+3 y^{2}$$

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• Paper 1, Section II, A

  #Numerical Analysis

  2010

  (a) State the Householder-John theorem and explain its relation to the convergence analysis of splitting methods for solving a system of linear equations $A x=b$ with a positive definite matrix $A$.

  (b) Describe the Jacobi method for solving a system $A x=b$, and deduce from the above theorem that if $A$ is a symmetric positive definite tridiagonal matrix,

  $$A=\left[\begin{array}{ccccc} a_{1} & c_{1} & & & \\ c_{1} & a_{2} & c_{2} & & \mathbf{0} \\ & \ddots & \ddots & \ddots & \\ \mathbf{0} & & c_{n-2} & a_{n-1} & c_{n-1} \\ & & & c_{n-1} & a_{n} \end{array}\right]$$

  then the Jacobi method converges.

  [Hint: At the last step, you may find it useful to consider two vectors $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ and $y=\left((-1) x_{1},(-1)^{2} x_{2}, \ldots,(-1)^{n} x_{n}\right)$.]

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• Paper 2, Section II, A

  #Numerical Analysis

  2010

  The inverse discrete Fourier transform $\mathcal{F}_{n}^{-1}: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is given by the formula

  $$\mathbf{x}=\mathcal{F}_{n}^{-1} \mathbf{y}, \quad \text { where } \quad x_{\ell}=\sum_{j=0}^{n-1} \omega_{n}^{j \ell} y_{j}, \quad \ell=0, \ldots, n-1$$

  Here, $\omega_{n}=\exp (2 \pi i / n)$ is the primitive root of unity of degree $n$, and $n=2^{p}, p=1,2, \ldots$

  (1) Show how to assemble $\mathbf{x}=\mathcal{F}_{2 m}^{-1} \mathbf{y}$ in a small number of operations if we already know the Fourier transforms of the even and odd portions of $\mathbf{y}$ :

  $$\mathbf{x}^{(\mathrm{E})}=\mathcal{F}_{m}^{-1} \mathbf{y}^{(\mathrm{E})}, \quad \mathbf{x}^{(\mathrm{O})}=\mathcal{F}_{m}^{-1} \mathbf{y}^{(\mathrm{O})}$$

  (2) Describe the Fast Fourier Transform (FFT) method for evaluating $\mathbf{x}$ and draw a relevant diagram for $n=8$.

  (3) Find the costs of the FFT for $n=2^{p}$ (only multiplications count).

  (4) For $n=4$, using the FFT technique, find $\mathbf{x}=\mathcal{F}_{4}^{-1} \mathbf{y}, \quad$ for $\quad \mathbf{y}=[1,1,-1,-1], \quad$ and $\quad \mathbf{y}=[1,-1,1,-1]$

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• Paper 3, Section II, A

  #Numerical Analysis

  2010

  The Poisson equation $\nabla^{2} u=f$ in the unit square $\Omega=[0,1] \times[0,1], u=0$ on $\partial \Omega$, is discretized with the five-point formula

  $$u_{i, j-1}+u_{i, j+1}+u_{i+1, j}+u_{i-1, j}-4 u_{i, j}=h^{2} f_{i, j}$$

  where $1 \leqslant i, j \leqslant M, u_{i, j} \approx u(i h, j h)$ and $(i h, j h)$ are grid points.

  Let $u(x, y)$ be the exact solution, and let $e_{i, j}=u_{i, j}-u(i h, j h)$ be the error of the five-point formula at the $(i, j)$ th grid point. Justifying each step, prove that

  $$\|\mathbf{e}\|=\left[\sum_{i, j=1}^{M}\left|e_{i, j}\right|^{2}\right]^{1 / 2} \leqslant c h \quad \text { for sufficiently small } h>0,$$

  where $c$ is some constant independent of $h$.

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• Paper 4, Section II, A

  #Numerical Analysis

  2010

  An $s$-stage explicit Runge-Kutta method of order $p$, with constant step size $h>0$, is applied to the differential equation $y^{\prime}=\lambda y, t \geqslant 0$.

  (a) Prove that

  $$y_{n+1}=P_{s}(\lambda h) y_{n}$$

  where $P_{s}$ is a polynomial of degree $s$.

  (b) Prove that the order $p$ of any $s$-stage explicit Runge-Kutta method satisfies the inequality $p \leqslant s$ and, for $p=s$, write down an explicit expression for $P_{s}$.

  (c) Prove that no explicit Runge-Kutta method can be A-stable.

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• Paper 2, Section II, J

  #Optimization and Control

  2010

  (a) Suppose that

  $$\left(\begin{array}{l} X \\ Y \end{array}\right) \sim N\left(\left(\begin{array}{l} \mu_{X} \\ \mu_{Y} \end{array}\right),\left(\begin{array}{ll} V_{X X} & V_{X Y} \\ V_{Y X} & V_{Y Y} \end{array}\right)\right)$$

  Prove that conditional on $Y=y$, the distribution of $X$ is again multivariate normal, with mean $\mu_{X}+V_{X Y} V_{Y Y}^{-1}\left(y-\mu_{Y}\right)$ and covariance $V_{X X}-V_{X Y} V_{Y Y}^{-1} V_{Y X}$.

  (b) The $\mathbb{R}^{d}$-valued process $X$ evolves in discrete time according to the dynamics

  $$X_{t+1}=A X_{t}+\varepsilon_{t+1}$$

  where $A$ is a constant $d \times d$ matrix, and $\varepsilon_{t}$ are independent, with common $N\left(0, \Sigma_{\varepsilon}\right)$ distribution. The process $X$ is not observed directly; instead, all that is seen is the process $Y$ defined as

  $$Y_{t}=C X_{t}+\eta_{t},$$

  where $\eta_{t}$ are independent of each other and of the $\varepsilon_{t}$, with common $N\left(0, \Sigma_{\eta}\right)$ distribution.

  If the observer has the prior distribution $X_{0} \sim N\left(\hat{X}_{0}, V_{0}\right)$ for $X_{0}$, prove that at all later times the distribution of $X_{t}$ conditional on $\mathcal{Y}_{t} \equiv\left(Y_{1}, \ldots, Y_{t}\right)$ is again normally distributed, with mean $\hat{X}_{t}$ and covariance $V_{t}$ which evolve as

  $$\begin{aligned} \hat{X}_{t+1} &=A \hat{X}_{t}+M_{t} C^{T}\left(\Sigma_{\eta}+C M_{t} C^{T}\right)^{-1}\left(Y_{t+1}-C A \hat{X}_{t}\right) \\ V_{t+1} &=M_{t}-M_{t} C^{T}\left(\Sigma_{\eta}+C M_{t} C^{T}\right)^{-1} C M_{t} \end{aligned}$$

  where

  $$M_{t}=A V_{t} A^{T}+\Sigma_{\varepsilon}$$

  (c) In the special case where both $X$ and $Y$ are one-dimensional, and $A=C=1$, $\Sigma_{\varepsilon}=0$, find the form of the updating recursion. Show in particular that

  $$\frac{1}{V_{t+1}}=\frac{1}{V_{t}}+\frac{1}{\Sigma_{\eta}}$$

  and that

  $$\frac{\hat{X}_{t+1}}{V_{t+1}}=\frac{\hat{X}_{t}}{V_{t}}+\frac{Y_{t+1}}{\Sigma_{\eta}}$$

  Hence deduce that, with probability one,

  $$\lim _{t \rightarrow \infty} \hat{X}_{t}=\lim _{t \rightarrow \infty} t^{-1} \sum_{j=1}^{t} Y_{j}$$

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• Paper 3, Section II, J

  #Optimization and Control

  2010

  Consider an infinite-horizon controlled Markov process having per-period costs $c(x, u) \geqslant 0$, where $x \in \mathcal{X}$ is the state of the system, and $u \in \mathcal{U}$ is the control. Costs are discounted at rate $\beta \in(0,1]$, so that the objective to be minimized is

  $$\mathbb{E}\left[\sum_{t \geqslant 0} \beta^{t} c\left(X_{t}, u_{t}\right) \mid X_{0}=x\right]$$

  What is meant by a policy $\pi$ for this problem?

  Let $\mathcal{L}$ denote the dynamic programming operator

  $$\mathcal{L} f(x) \equiv \inf _{u \in \mathcal{U}}\left\{c(x, u)+\beta \mathbb{E}\left[f\left(X_{1}\right) \mid X_{0}=x, u_{0}=u\right]\right\}$$

  Further, let $F$ denote the value of the optimal control problem:

  $$F(x)=\inf _{\pi} \mathbb{E}^{\pi}\left[\sum_{t \geqslant 0} \beta^{t} c\left(X_{t}, u_{t}\right) \mid X_{0}=x\right]$$

  where the infimum is taken over all policies $\pi$, and $\mathbb{E}^{\pi}$ denotes expectation under policy $\pi$. Show that the functions $F_{t}$ defined by

  $$F_{t+1}=\mathcal{L} F_{t} \quad(t \geqslant 0), \quad F_{0} \equiv 0$$

  increase to a limit $F_{\infty} \in[0, \infty] .$ Prove that $F_{\infty} \leqslant F$. Prove that $F=\mathcal{L} F$

  Suppose that $\Phi=\mathcal{L} \Phi \geqslant 0$. Prove that $\Phi \geqslant F$.

  [You may assume that there is a function $u_{*}: \mathcal{X} \rightarrow \mathcal{U}$ such that

  $$\mathcal{L} \Phi(x)=c\left(x, u_{*}(x)\right)+\beta \mathbb{E}\left[\Phi\left(X_{1}\right) \mid X_{0}=x, u_{0}=u_{*}(x)\right]$$

  though the result remains true without this simplifying assumption.]

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• Paper 4, Section II, J

  #Optimization and Control

  2010

  Dr Seuss' wealth $x_{t}$ at time $t$ evolves as

  $$\frac{d x}{d t}=r x_{t}+\ell_{t}-c_{t}$$

  where $r>0$ is the rate of interest earned, $\ell_{t}$ is his intensity of working $(0 \leqslant \ell \leqslant 1)$, and $c_{t}$ is his rate of consumption. His initial wealth $x_{0}>0$ is given, and his objective is to maximize

  $$\int_{0}^{T} U\left(c_{t}, \ell_{t}\right) d t$$

  where $U(c, \ell)=c^{\alpha}(1-\ell)^{\beta}$, and $T$ is the (fixed) time his contract expires. The constants $\alpha$ and $\beta$ satisfy the inequalities $0<\alpha<1,0<\beta<1$, and $\alpha+\beta>1$. At all times, $c_{t}$ must be non-negative, and his final wealth $x_{T}$ must be non-negative. Establish the following properties of the optimal solution $\left(x^{*}, c^{*}, \ell^{*}\right)$ :

  (i) $\beta c_{t}^{*}=\alpha\left(1-\ell_{t}^{*}\right)$;

  (ii) $c_{t}^{*} \propto e^{-\gamma r t}$, where $\gamma \equiv(\beta-1+\alpha)^{-1}$;

  (iii) $x_{t}^{*}=A e^{r t}+B e^{-\gamma r t}-r^{-1}$ for some constants $A$ and $B$.

  Hence deduce that the optimal wealth is

  $$x_{t}^{*}=\frac{\left(1-e^{-\gamma r T}\left(1+r x_{0}\right)\right) e^{r t}+\left(\left(1+r x_{0}\right) e^{r T}-1\right) e^{-\gamma r t}}{r\left(e^{r T}-e^{-\gamma r T}\right)}-\frac{1}{r}$$

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• Paper 1, Section II, E

  #Partial Differential Equations

  2010

  (a) Solve by using the method of characteristics

  $$x_{1} \frac{\partial}{\partial x_{1}} u+2 x_{2} \frac{\partial}{\partial x_{2}} u=5 u, \quad u\left(x_{1}, 1\right)=g\left(x_{1}\right),$$

  where $g: \mathbb{R} \rightarrow \mathbb{R}$ is continuous. What is the maximal domain in $\mathbb{R}^{2}$ in which $u$ is a solution of the Cauchy problem?

  (b) Prove that the function

  $$u(x, t)=\left\{\begin{array}{cl} 0, & x<0, t>0 \\ x / t, & 0<x<t, t>0 \\ 1, & x>t>0 \end{array}\right.$$

  is a weak solution of the Burgers equation

  $$\frac{\partial}{\partial t} u+\frac{1}{2} \frac{\partial}{\partial x} u^{2}=0, \quad x \in \mathbb{R}, t>0$$

  with initial data

  $$u(x, 0)= \begin{cases}0, & x<0 \\ 1, & x>0\end{cases}$$

  (c) Let $u=u(x, t), x \in \mathbb{R}, t>0$ be a piecewise $C^{1}$-function with a jump discontinuity along the curve

  $$\Gamma: x=s(t)$$

  and let $u$ solve the Burgers equation $(*)$ on both sides of $\Gamma$. Prove that $u$ is a weak solution of (1) if and only if

  $$\dot{s}(t)=\frac{1}{2}\left(u_{l}(t)+u_{r}(t)\right)$$

  holds, where $u_{l}(t), u_{r}(t)$ are the one-sided limits

  $$u_{l}(t)=\lim _{x \nearrow_{s}(t)^{-}} u(x, t), \quad u_{r}(t)=\lim _{x \searrow s(t)^{+}} u(x, t)$$

  [Hint: Multiply the equation by a test function $\phi \in C_{0}^{\infty}(\mathbb{R} \times[0, \infty))$, split the integral appropriately and integrate by parts. Consider how the unit normal vector along $\Gamma$ can be expressed in terms of $\dot{s}$.]

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• Paper 2, Section II, E

  #Partial Differential Equations

  2010

  (a) State the Lax-Milgram lemma. Use it to prove that there exists a unique function $u$ in the space

  $$H_{\partial}^{2}(\Omega)=\left\{u \in H^{2}(\Omega) ;\left.u\right|_{\partial \Omega}=\partial u /\left.\partial \gamma\right|_{\partial \Omega}=0\right\}$$

  where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary and $\gamma$ its outwards unit normal vector, which is the weak solution of the equations

  $$\begin{aligned} \Delta^{2} u &=f \text { in } \Omega, \\ u &=\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega, \end{aligned}$$

  for $f \in L^{2}(\Omega), \Delta$ the Laplacian and $\Delta^{2}=\Delta \Delta$.

  [Hint: Use regularity of the solution of the Dirichlet problem for the Poisson equation.]

  (b) Let $\Omega \subset \mathbb{R}^{n}$ be a bounded domain with smooth boundary. Let $u \in H^{1}(\Omega)$ and denote

  $$\bar{u}=\int_{\Omega} u d^{n} x / \int_{\Omega} d^{n} x$$

  The following Poincaré-type inequality is known to hold

  $$\|u-\bar{u}\|_{L^{2}} \leqslant C\|\nabla u\|_{L^{2}}$$

  where $C$ only depends on $\Omega$. Use the Lax-Milgram lemma and this Poincaré-type inequality to prove that the Neumann problem

  $$\begin{aligned} &\Delta u=f \text { in } \Omega \\ &\frac{\partial u}{\partial \gamma}=0 \text { on } \partial \Omega \end{aligned}$$

  has a unique weak solution in the space

  $$H_{-}^{1}(\Omega)=H^{1}(\Omega) \cap\{u: \Omega \rightarrow \mathbb{R} ; \bar{u}=0\}$$

  if and only if $\bar{f}=0$.

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• Paper 3, Section II, 30E

  #Partial Differential Equations

  2010

  Consider the Schrödinger equation

  $$i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, \quad x \in \mathbb{R}^{n}, t>0$$

  for complex-valued solutions $\Psi(x, t)$ and where $\Delta$ is the Laplacian.

  (a) Derive, by using a Fourier transform and its inversion, the fundamental solution of the Schrödinger equation. Obtain the solution of the initial value problem

  $$\begin{aligned} i \partial_{t} \Psi=-\frac{1}{2} \Delta \Psi, & x \in \mathbb{R}^{n}, \quad t>0 \\ \Psi(x, 0)=f(x), & x \in \mathbb{R}^{n} \end{aligned}$$

  as a convolution.

  (b) Consider the Wigner-transform of the solution of the Schrödinger equation

  $$w(x, \xi, t)=\frac{1}{(2 \pi)^{n}} \int_{\mathbb{R}^{n}} \Psi\left(x+\frac{1}{2} y, t\right) \bar{\Psi}\left(x-\frac{1}{2} y, t\right) e^{-i y \cdot \xi} \mathrm{d}^{n} y$$

  defined for $x \in \mathbb{R}^{n}, \xi \in \mathbb{R}^{n}, t>0$. Derive an evolution equation for $w$ by using the Schrödinger equation. Write down the solution of this evolution equation for given initial data $w(x, \xi, 0)=g(x, \xi)$.

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• Paper 4, Section II, 30E

  #Partial Differential Equations

  2010

  a) Solve the Dirichlet problem for the Laplace equation in a disc in $\mathbb{R}^{2}$

  $$\begin{aligned} \Delta u &=0 \quad \text { in } \quad G=\left\{x^{2}+y^{2}<R^{2}\right\} \subseteq \mathbb{R}^{2}, R>0 \\ u &=u_{D} \quad \text { on } \quad \partial G \end{aligned}$$

  using polar coordinates $(r, \varphi)$ and separation of variables, $u(x, y)=R(r) \Theta(\varphi)$. Then use the ansatz $R(r)=r^{\alpha}$ for the radial function.

  b) Solve the Dirichlet problem for the Laplace equation in a square in $\mathbb{R}^{2}$

  $$\begin{aligned} &\Delta u=0 \quad \text { in } \quad G=[0, a] \times[0, a] \\ &u(x, 0)=f_{1}(x), \quad u(x, a)=f_{2}(x), \quad u(0, y)=f_{3}(y), \quad u(a, y)=f_{4}(y) \end{aligned}$$

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• Paper 1, Section II, C

  #Principles of Quantum Mechanics

  2010

  Two states $\left|j_{1} m_{1}\right\rangle_{1},\left|j_{2} m_{2}\right\rangle_{2}$, with angular momenta $j_{1}, j_{2}$, are combined to form states $|J M\rangle$ with total angular momentum

  $$J=\left|j_{1}-j_{2}\right|,\left|j_{1}-j_{2}\right|+1, \ldots, j_{1}+j_{2} .$$

  Write down the state with $J=M=j_{1}+j_{2}$ in terms of the original angular momentum states. Briefly describe how the other combined angular momentum states may be found in terms of the original angular momentum states.

  If $j_{1}=j_{2}=j$, explain why the state with $J=0$ must be of the form

  $$|00\rangle=\sum_{m=-j}^{j} \alpha_{m}|j m\rangle_{1}|j-m\rangle_{2}$$

  By considering $J_{+}|00\rangle$, determine a relation between $\alpha_{m+1}$ and $\alpha_{m}$, hence find $\alpha_{m}$.

  If the system is in the state $|j j\rangle_{1}|j-j\rangle_{2}$ what is the probability, written in terms of $j$, of measuring the combined total angular momentum to bero?

  [Standard angular momentum states $|j m\rangle$ are joint eigenstates of $\mathbf{J}^{2}$ and $J_{3}$, obeying

  $$J_{\pm}|j m\rangle=\sqrt{(j \mp m)(j \pm m+1)}|j m \pm 1\rangle$$

  Units in which $\hbar=1$ have been used throughout.]

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• Paper 2, Section II, C

  #Principles of Quantum Mechanics

  2010

  Consider a joint eigenstate of $\mathbf{J}^{2}$ and $J_{3},|j m\rangle$. Write down a unitary operator $U(\mathbf{n}, \theta)$ for rotation of the state by an angle $\theta$ about an axis with direction $\mathbf{n}$, where $\mathbf{n}$ is a unit vector. How would a state with zero orbital angular momentum transform under such a rotation?

  What is the relation between the angular momentum operator $\mathbf{J}$ and the Pauli matrices $\boldsymbol{\sigma}$ when $j=\frac{1}{2}$ ? Explicitly calculate $(\mathbf{J} \cdot \mathbf{a})^{2}$, for an arbitrary real vector $\mathbf{a}$, in this case. What are the eigenvalues of the operator $\mathbf{J} \cdot \mathbf{a}$ ? Show that the unitary rotation operator for $j=\frac{1}{2}$ can be expressed as

  $$U(\mathbf{n}, \theta)=\cos \frac{\theta}{2}-i \mathbf{n} \cdot \boldsymbol{\sigma} \sin \frac{\theta}{2}$$

  Starting with a state $\left|\frac{1}{2} m\right\rangle$ the component of angular momentum along a direction $\mathbf{n}^{\prime}$, making and angle $\theta$ with the $z$-axis, is susequently measured to be $m^{\prime}$. Immediately after this measurement the state is $\left|\frac{1}{2} m^{\prime}\right\rangle_{\theta}$. Write down an eigenvalue equation for $\left|\frac{1}{2} m^{\prime}\right\rangle_{\theta}$ in terms of $\mathbf{n}^{\prime} \cdot \mathbf{J}$. Show that the probability for measuring an angular momentum of $m^{\prime} \hbar$ along the direction $\mathbf{n}^{\prime}$ is, assuming $\mathbf{n}^{\prime}$ is in the $x-z$ plane,

  $$\left|\left\langle\frac{1}{2} m \mid \frac{1}{2} m^{\prime}\right\rangle_{\theta}\right|^{2}=\left|\left\langle\frac{1}{2} m|U(\mathbf{y}, \theta)| \frac{1}{2} m^{\prime}\right\rangle\right|^{2},$$

  where $\mathbf{y}$ is a unit vector in the $y$-direction. Using $(*)$ show that the probability that $m=+\frac{1}{2}, m^{\prime}=-\frac{1}{2}$ is of the form

  $$A+B \cos ^{2} \frac{\theta}{2}$$

  determining the integers $A$ and $B$ in the process.

  [Assume $\hbar=1$. The Pauli matrices are

  $$\sigma_{1}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$$

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• Paper 3, Section II, C

  #Principles of Quantum Mechanics

  2010

  What are the commutation relations between the position operator $\hat{x}$ and momentum operator $\hat{p}$ ? Show that this is consistent with $\hat{x}, \hat{p}$ being hermitian.

  The annihilation operator for a harmonic oscillator is

  $$a=\sqrt{\frac{1}{2 \hbar}}(\hat{x}+i \hat{p})$$

  in units where the mass and frequency of the oscillator are 1 . Derive the relation $\left[a, a^{\dagger}\right]=1$. Write down an expression for the Hamiltonian

  $$H=\frac{1}{2} \hat{p}^{2}+\frac{1}{2} \hat{x}^{2}$$

  in terms of the operator $N=a^{\dagger} a$.

  Assume there exists a unique ground state $|0\rangle$ of $H$ such that $a|0\rangle=0$. Explain how the space of eigenstates $|n\rangle$, is formed, and deduce the energy eigenvalues for these states. Show that

  $$a|n\rangle=A|n-1\rangle, \quad a^{\dagger}|n\rangle=B|n+1\rangle$$

  finding $A$ and $B$ in terms of $n$.

  Calculate the energy eigenvalues of the Hamiltonian for two harmonic oscillators

  $$H=H_{1}+H_{2}, \quad H_{i}=\frac{1}{2} \hat{p}_{i}^{2}+\frac{1}{2} \hat{x}_{i}^{2}, \quad i=1,2 .$$

  What is the degeneracy of the $n^{\text {th }}$ energy level? Suppose that the two oscillators are then coupled by adding the extra term

  $$\Delta H=\lambda \hat{x}_{1} \hat{x}_{2}$$

  to $H$, where $\lambda \ll 1$. Calculate the energies for the states of the unperturbed system with the three lowest energy eigenvalues to first order in $\lambda$ using perturbation theory.

  [You may assume standard perturbation theory results.]

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• Paper 4, Section II, C

  #Principles of Quantum Mechanics

  2010

  The Hamiltonian for a quantum system in the Schrödinger picture is

  $$H_{0}+V(t)$$

  where $H_{0}$ is independent of time. Define the interaction picture corresponding to this Hamiltonian and derive a time evolution equation for interaction picture states.

  Let $|a\rangle$ and $|b\rangle$ be orthonormal eigenstates of $H_{0}$ with eigenvalues $E_{a}$ and $E_{b}$ respectively. Assume $V(t)=0$ for $t \leqslant 0$. Show that if the system is initially, at $t=0$, in the state $|a\rangle$ then the probability of measuring it to be the state $|b\rangle$ after a time $t$ is

  $$\frac{1}{\hbar^{2}}\left|\int_{0}^{t} d t^{\prime}\left\langle b\left|V\left(t^{\prime}\right)\right| a\right\rangle e^{i\left(E_{b}-E_{a}\right) t^{\prime} / \hbar}\right|^{2}$$

  to order $V(t)^{2}$.

  Suppose a system has a basis of just two orthonormal states $|1\rangle$ and $|2\rangle$, with respect to which

  $$H_{0}=E I, \quad V(t)=v t \sigma_{1}, \quad t \geqslant 0,$$

  where

  $$I=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right), \quad \sigma_{1}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)$$

  Use $(*)$ to calculate the probability of a transition from state $|1\rangle$ to state $|2\rangle$ after a time $t$ to order $v^{2}$.

  Show that the time dependent Schrödinger equation has a solution

  $$|\psi(t)\rangle=\exp \left(-\frac{i}{\hbar}\left(E t I+\frac{1}{2} v t^{2} \sigma_{1}\right)\right)|\psi(0)\rangle$$

  Calculate the transition probability exactly. Hence find the condition for the order $v^{2}$ approximation to be valid.

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• Paper 1, Section II, J

  #Principles of Statistics

  2010

  The distribution of a random variable $X$ is obtained from the binomial distribution $\mathcal{B}(n ; \Pi)$ by conditioning on $X>0$; here $\Pi \in(0,1)$ is an unknown probability parameter and $n$ is known. Show that the distributions of $X$ form an exponential family and identify the natural sufficient statistic $T$, natural parameter $\Phi$, and cumulant function $k(\phi)$. Using general properties of the cumulant function, compute the mean and variance of $X$ when $\Pi=\pi$. Write down an equation for the maximum likelihood estimate $\widehat{\Pi}$ of $\Pi$ and explain why, when $\Pi=\pi$, the distribution of $\widehat{\Pi}$ is approximately normal $\mathcal{N}(\pi, \pi(1-\pi) / n)$ for large $n$.

  Suppose we observe $X=1$. It is suggested that, since the condition $X>0$ is then automatically satisfied, general principles of inference require that the inference to be drawn should be the same as if the distribution of $X$ had been $\mathcal{B}(n ; \Pi)$ and we had observed $X=1$. Comment briefly on this suggestion.

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• Paper 2, Section II, J

  #Principles of Statistics

  2010

  Define the Kolmogorov-Smirnov statistic for testing the null hypothesis that real random variables $X_{1}, \ldots, X_{n}$ are independently and identically distributed with specified continuous, strictly increasing distribution function $F$, and show that its null distribution does not depend on $F$.

  A composite hypothesis $H_{0}$ specifies that, when the unknown positive parameter $\Theta$ takes value $\theta$, the random variables $X_{1}, \ldots, X_{n}$ arise independently from the uniform distribution $\mathrm{U}[0, \theta]$. Letting $J:=\arg \max _{1 \leqslant i \leqslant n} X_{i}$, show that, under $H_{0}$, the statistic $\left(J, X_{J}\right)$ is sufficient for $\Theta$. Show further that, given $\left\{J=j, X_{j}=\xi\right\}$, the random variables $\left(X_{i}: i \neq j\right)$ are independent and have the $\mathrm{U}[0, \xi]$ distribution. How might you apply the Kolmogorov-Smirnov test to test the hypothesis $H_{0}$ ?

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• Paper 3, Section II, $\mathbf{2 7 J}$

  #Principles of Statistics

  2010

  Define the normal and extensive form solutions of a Bayesian statistical decision problem involving parameter $\Theta$, random variable $X$, and loss function $L(\theta, a)$. How are they related? Let $R_{0}=R_{0}(\Pi)$ be the Bayes loss of the optimal act when $\Theta \sim \Pi$ and no data can be observed. Express the Bayes risk $R_{1}$ of the optimal statistical decision rule in terms of $R_{0}$ and the joint distribution of $(\Theta, X)$.

  The real parameter $\Theta$ has distribution $\Pi$, having probability density function $\pi(\cdot)$. Consider the problem of specifying a set $S \subseteq \mathbb{R}$ such that the loss when $\Theta=\theta$ is $L(\theta, S)=c|S|-\mathbf{1}_{S}(\theta)$, where $\mathbf{1}_{S}$ is the indicator function of $S$, where $c>0$, and where $|S|=\int_{S} d x$. Show that the "highest density" region $S^{*}:=\{\theta: \pi(\theta) \geqslant c\}$ supplies a Bayes act for this decision problem, and explain why $R_{0}(\Pi) \leqslant 0$.

  For the case $\Theta \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$, find an expression for $R_{0}$ in terms of the standard normal distribution function $\Phi$.

  Suppose now that $c=0.5$, that $\Theta \sim \mathcal{N}(0,1)$ and that $X \mid \Theta \sim \mathcal{N}(\Theta, 1 / 9)$. Show that $R_{1}<R_{0}$.

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• Paper 4, Section II, $\mathbf{2 7 J}$

  #Principles of Statistics

  2010

  Define completeness and bounded completeness of a statistic $T$ in a statistical experiment.

  Random variables $X_{1}, X_{2}, X_{3}$ are generated as $X_{i}=\Theta^{1 / 2} Z+(1-\Theta)^{1 / 2} Y_{i}$, where $Z, Y_{1}, Y_{2}, Y_{3}$ are independently standard normal $\mathcal{N}(0,1)$, and the parameter $\Theta$ takes values in $(0,1)$. What is the joint distribution of $\left(X_{1}, X_{2}, X_{3}\right)$ when $\Theta=\theta$ ? Write down its density function, and show that a minimal sufficient statistic for $\Theta$ based on $\left(X_{1}, X_{2}, X_{3}\right)$ is $T=\left(T_{1}, T_{2}\right):=\left(\sum_{i=1}^{3} X_{i}^{2},\left(\sum_{i=1}^{3} X_{i}\right)^{2}\right)$.

  [Hint: You may use that if $I$ is the $n \times n$ identity matrix and $J$ is the $n \times n$ matrix all of whose entries are 1 , then $a I+b J$ has determinant $a^{n-1}(a+n b)$, and inverse $c I+d J$ with $c=1 / a, d=-b /(a(a+n b))$.]

  What is $\mathbb{E}_{\theta}\left(T_{1}\right) ?$ Is $T$ complete for $\Theta ?$

  Let $S:=\operatorname{Prob}\left(X_{1}^{2} \leqslant 1 \mid T\right)$. Show that $\mathbb{E}_{\theta}(S)$ is a positive constant $c$ which does not depend on $\theta$, but that $S$ is not identically equal to $c$. Is $T$ boundedly complete for $\Theta$ ?

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• Paper 1, Section II, I

  #Probability and Measure

  2010

  State Carathéodory's extension theorem. Define all terms used in the statement.

  Let $\mathcal{A}$ be the ring of finite unions of disjoint bounded intervals of the form

  $$A=\bigcup_{i=1}^{m}\left(a_{i}, b_{i}\right]$$

  where $m \in \mathbb{Z}^{+}$and $a_{1}<b_{1}<\ldots<a_{m}<b_{m}$. Consider the set function $\mu$ defined on $\mathcal{A}$ by

  $$\mu(A)=\sum_{i=1}^{m}\left(b_{i}-a_{i}\right)$$

  You may assume that $\mu$ is additive. Show that for any decreasing sequence $\left(B_{n}: n \in \mathbb{N}\right)$ in $\mathcal{A}$ with empty intersection we have $\mu\left(B_{n}\right) \rightarrow 0$ as $n \rightarrow \infty$.

  Explain how this fact can be used in conjunction with Carathéodory's extension theorem to prove the existence of Lebesgue measure.

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• Paper 2, Section II, I

  #Probability and Measure

  2010

  Show that any two probability measures which agree on a $\pi$-system also agree on the $\sigma$-algebra generated by that $\pi$-system.

  State Fubini's theorem for non-negative measurable functions.

  Let $\mu$ denote Lebesgue measure on $\mathbb{R}^{2}$. Fix $s \in[0,1)$. Set $c=\sqrt{1-s^{2}}$ and $\lambda=\sqrt{c}$. Consider the linear maps $f, g, h: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ given by

  $$f(x, y)=\left(\lambda^{-1} x, \lambda y\right), \quad g(x, y)=(x, s x+y), \quad h(x, y)=(x-s y, y)$$

  Show that $\mu=\mu \circ f^{-1}$ and that $\mu=\mu \circ g^{-1}$. You must justify any assertion you make concerning the values taken by $\mu$.

  Compute $r=f \circ h \circ g \circ f$. Deduce that $\mu$ is invariant under rotations.

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• Paper 3, Section II, I

  #Probability and Measure

  2010

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables with common density function

  $$f(x)=\frac{1}{\pi\left(1+x^{2}\right)}$$

  Fix $\alpha \in[0,1]$ and set

  $$Y_{n}=\operatorname{sgn}\left(X_{n}\right)\left|X_{n}\right|^{\alpha}, \quad S_{n}=Y_{1}+\ldots+Y_{n}$$

  Show that for all $\alpha \in[0,1]$ the sequence of random variables $S_{n} / n$ converges in distribution and determine the limit.

  [Hint: In the case $\alpha=1$ it may be useful to prove that $\mathbb{E}\left(e^{i u X_{1}}\right)=e^{-|u|}$, for all $\left.u \in \mathbb{R} .\right]$

  Show further that for all $\alpha \in[0,1 / 2)$ the sequence of random variables $S_{n} / \sqrt{n}$ converges in distribution and determine the limit.

  [You should state clearly any result about random variables from the course to which you appeal. You are not expected to evaluate explicitly the integral

  $$m(\alpha)=\int_{0}^{\infty} \frac{x^{\alpha}}{\pi\left(1+x^{2}\right)} d x$$

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• Paper 4, Section II, I

  #Probability and Measure

  2010

  Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent normal random variables having mean 0 and variance 1 . Set $S_{n}=X_{1}+\ldots+X_{n}$ and $U_{n}=S_{n}-\left\lfloor S_{n}\right\rfloor$. Thus $U_{n}$ is the fractional part of $S_{n}$. Show that $U_{n}$ converges to $U$ in distribution, as $n \rightarrow \infty$ where $U$ is uniformly distributed on $[0,1]$.

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• Paper 1, Section II, F

  #Representation Theory

  2010

  (i) Let $N$ be a normal subgroup of the finite group $G$. Without giving detailed proofs, define the process of lifting characters from $G / N$ to $G$. State also the orthogonality relations for $G$.

  (ii) Let $a, b$ be the following two permutations in $S_{12}$,

  and let $G=\langle a, b\rangle$, a subgroup of $S_{12}$. Prove that $G$ is a group of order 12 and list the conjugacy classes of $G$. By identifying a normal subgroup of $G$ of index 4 and lifting irreducible characters, calculate all the linear characters of $G$. Calculate the complete character table of $G$. By considering 6 th roots of unity, find explicit matrix representations affording the non-linear characters of $G$.

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• Paper 2, Section II, F

  #Representation Theory

  2010

  Define the concepts of induction and restriction of characters. State and prove the Frobenius Reciprocity Theorem.

  Let $H$ be a subgroup of $G$ and let $g \in G$. We write $\mathcal{C}(g)$ for the conjugacy class of $g$ in $G$, and write $C_{G}(g)$ for the centraliser of $g$ in $G$. Suppose that $H \cap \mathcal{C}(g)$ breaks up into $m$ conjugacy classes of $H$, with representatives $x_{1}, x_{2}, \ldots, x_{m}$.

  Let $\psi$ be a character of $H$. Writing $\operatorname{Ind}_{H}^{G}(\psi)$ for the induced character, prove that

  (i) if no element of $\mathcal{C}(g)$ lies in $H$, then $\operatorname{Ind}_{H}^{G}(\psi)(g)=0$,

  (ii) if some element of $\mathcal{C}(g)$ lies in $H$, then

  $$\operatorname{Ind}_{H}^{G}(\psi)(g)=\left|C_{G}(g)\right| \sum_{i=1}^{m} \frac{\psi\left(x_{i}\right)}{\left|C_{H}\left(x_{i}\right)\right|} .$$

  Let $G=S_{4}$ and let $H=\langle a, b\rangle$, where $a=\left(\begin{array}{llll}1 & 2 & 3 & 4\end{array}\right)$ and $b=\left(\begin{array}{l}1\end{array}\right.$ dihedral group and write down its character table. Restrict each $G$-conjugacy class to $H$ and calculate the $H$-conjugacy classes contained in each restriction. Given a character $\psi$ of $H$, express Ind ${ }_{H}^{G}(\psi)(g)$ in terms of $\psi$, where $g$ runs through a set of conjugacy classes of $G$. Use your calculation to find the values of all the irreducible characters of $H$ induced to $G$.

  $$\begin{aligned} & a=\left(\begin{array}{lllll}1 & 2 & 3 & 5 & 6\end{array}\right)(789101112), \\ & b=\left(\begin{array}{llll}1 & 7 & 4 & 10\end{array}\right)\left(\begin{array}{llll}2 & 12 & 5 & 9\end{array}\right)(368), \end{aligned}$$

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• Paper 3, Section II, F

  #Representation Theory

  2010

  Show that the degree of a complex irreducible character of a finite group is a factor of the order of the group.

  State and prove Burnside's $p^{a} q^{b}$ theorem. You should quote clearly any results you use.

  Prove that for any group of odd order $n$ having precisely $k$ conjugacy classes, the integer $n-k$ is divisible by 16 .

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• Paper 4, Section II, F

  #Representation Theory

  2010

  Define the circle group $U(1)$. Give a complete list of the irreducible representations of $U(1)$

  Define the spin group $G=S U(2)$, and explain briefly why it is homeomorphic to the unit 3-sphere in $\mathbb{R}^{4}$. Identify the conjugacy classes of $G$ and describe the classification of the irreducible representations of $G$. Identify the characters afforded by the irreducible representations. You need not give detailed proofs but you should define all the terms you use.

  Let $G$ act on the space $\mathrm{M}_{3}(\mathbb{C})$ of $3 \times 3$ complex matrices by conjugation, where $A \in S U(2)$ acts by

  $$A: M \mapsto A_{1} M A_{1}^{-1}$$

  in which $A_{1}$ denotes the $3 \times 3$ block diagonal matrix $\left(\begin{array}{cc}A & 0 \\ 0 & 1\end{array}\right)$. Show that this gives a representation of $G$ and decompose it into irreducibles.

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• Paper 1, Section II, G

  #Riemann Surfaces

  2010

  Given a lattice $\Lambda \subset \mathbb{C}$, we may define the corresponding Weierstrass $\wp$-function to be the unique even $\Lambda$-periodic elliptic function $\wp$ with poles only on $\Lambda$ and for which $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. For $w \notin \Lambda$, we set

  $$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(w) & \wp(-z-w) \\ \wp^{\prime}(z) & \wp^{\prime}(w) & \wp^{\prime}(-z-w) \end{array}\right)$$

  an elliptic function with periods $\Lambda$. By considering the poles of $f$, show that $f$ has valency at most 4 (i.e. is at most 4 to 1 on a period parallelogram).

  If $w \notin \frac{1}{3} \Lambda$, show that $f$ has at least six distinct zeros. If $w \in \frac{1}{3} \Lambda$, show that $f$ has at least four distinct zeros, at least one of which is a multiple zero. Deduce that the meromorphic function $f$ is identically zero.

  If $z_{1}, z_{2}, z_{3}$ are distinct non-lattice points in a period parallelogram such that $z_{1}+z_{2}+z_{3} \in \Lambda$, what can be said about the points $\left(\wp\left(z_{i}\right), \wp^{\prime}\left(z_{i}\right)\right) \in \mathbb{C}^{2}(i=1,2,3) ?$

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• Paper 2, Section II, G

  #Riemann Surfaces

  2010

  Given a complete analytic function $\mathcal{F}$ on a domain $U \subset \mathbb{C}$, describe briefly how the space of germs construction yields a Riemann surface $R$ associated to $\mathcal{F}$ together with a covering map $\pi: R \rightarrow U$ (proofs not required).

  In the case when $\pi$ is regular, explain briefly how, given a point $P \in U$, any closed curve in $U$ with initial and final points $P$ yields a permutation of the set $\pi^{-1}(P)$.

  Now consider the Riemann surface $R$ associated with the complete analytic function

  $$\left(z^{2}-1\right)^{1 / 2}+\left(z^{2}-4\right)^{1 / 2}$$

  on $U=\mathbb{C} \backslash\{\pm 1, \pm 2\}$, with regular covering map $\pi: R \rightarrow U$. Which subgroup of the full symmetric group of $\pi^{-1}(P)$ is obtained in this way from all such closed curves (with initial and final points $P)$ ?

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• Paper 3, Section II, G

  #Riemann Surfaces

  2010

  Show that the analytic isomorphisms (i.e. conformal equivalences) of the Riemann sphere $\mathbb{C}_{\infty}$ to itself are given by the non-constant Möbius transformations.

  State the Riemann-Hurwitz formula for a non-constant analytic map between compact Riemann surfaces, carefully explaining the terms which occur.

  Suppose now that $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is an analytic map of degree 2 ; show that there exist Möbius transformations $S$ and $T$ such that

  $$S f T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$$

  is the map given by $z \mapsto z^{2}$.

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• Paper 1, Section I, J

  #Statistical Modelling

  2010

  Consider a binomial generalised linear model for data $y_{1}, \ldots, y_{n}$ modelled as realisations of independent $Y_{i} \sim \operatorname{Bin}\left(1, \mu_{i}\right)$ and logit $\operatorname{link} \mu_{i}=e^{\beta x_{i}} /\left(1+e^{\beta x_{i}}\right)$ for some known constants $x_{i}, i=1, \ldots, n$, and unknown scalar parameter $\beta$. Find the log-likelihood for $\beta$, and the likelihood equation that must be solved to find the maximum likelihood estimator $\hat{\beta}$ of $\beta$. Compute the second derivative of the log-likelihood for $\beta$, and explain the algorithm you would use to find $\hat{\beta}$.

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• Paper 2, Section I, J

  #Statistical Modelling

  2010

  Suppose you have a parametric model consisting of probability mass functions $f(y ; \theta), \theta \in \Theta \subset \mathbb{R}$. Given a sample $Y_{1}, \ldots, Y_{n}$ from $f(y ; \theta)$, define the maximum likelihood estimator $\hat{\theta}_{n}$ for $\theta$ and, assuming standard regularity conditions hold, state the asymptotic distribution of $\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)$.

  Compute the Fisher information of a single observation in the case where $f(y ; \theta)$ is the probability mass function of a Poisson random variable with parameter $\theta$. If $Y_{1}, \ldots, Y_{n}$ are independent and identically distributed random variables having a Poisson distribution with parameter $\theta$, show that $\bar{Y}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}$ and $S=\frac{1}{n-1} \sum_{i=1}^{n}\left(Y_{i}-\bar{Y}\right)^{2}$ are unbiased estimators for $\theta$. Without calculating the variance of $S$, show that there is no reason to prefer $S$ over $\bar{Y}$.

  [You may use the fact that the asymptotic variance of $\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)$ is a lower bound for the variance of any unbiased estimator.]

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• Paper 3, Section I, J

  #Statistical Modelling

  2010

  Consider the linear model $Y=X \beta+\varepsilon$, where $Y$ is a $n \times 1$ random vector, $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$, and where the $n \times p$ nonrandom matrix $X$ is known and has full column rank $p$. Derive the maximum likelihood estimator $\hat{\sigma}^{2}$ of $\sigma^{2}$. Without using Cochran's theorem, show carefully that $\hat{\sigma}^{2}$ is biased. Suggest another estimator $\tilde{\sigma}^{2}$ for $\sigma^{2}$ that is unbiased.

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• Paper 4, Section I, J

  #Statistical Modelling

  2010

  Below is a simplified 1993 dataset of US cars. The columns list index, make, model, price (in $\$ 1000)$, miles per gallon, number of passengers, length and width in inches, and weight (in pounds). The data are displayed in $R$ as follows (abbreviated):

  

  It is reasonable to assume that prices for different makes of car are independent. We model the logarithm of the price as a linear combination of the other quantitative properties of the cars and an error term. Write down this model mathematically. How would you instruct $R$ to fit this model and assign it to a variable "fit"?

  $R$ provides the following (slightly abbreviated) summary:

  

  Briefly explain the information that is being provided in each column of the table. What are your conclusions and how would you try to improve the model?

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• Paper 1, Section II, J

  #Statistical Modelling

  2010

  Consider a generalised linear model with parameter $\beta^{\top}$ partitioned as $\left(\beta_{0}^{\top}, \beta_{1}^{\top}\right)$, where $\beta_{0}$ has $p_{0}$ components and $\beta_{1}$ has $p-p_{0}$ components, and consider testing $H_{0}: \beta_{1}=0$ against $H_{1}: \beta_{1} \neq 0$. Define carefully the deviance, and use it to construct a test for $H_{0}$.

  [You may use Wilks' theorem to justify this test, and you may also assume that the dispersion parameter is known.]

  Now consider the generalised linear model with Poisson responses and the canonical link function with linear predictor $\eta=\left(\eta_{1}, \ldots, \eta_{n}\right)^{T}$ given by $\eta_{i}=x_{i}^{\top} \beta, i=1, \ldots, n$, where $x_{i 1}=1$ for every $i$. Derive the deviance for this model, and argue that it may be approximated by Pearson's $\chi^{2}$ statistic.

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• Paper 4, Section II, J

  #Statistical Modelling

  2010

  Every day, Barney the darts player comes to our laboratory. We record his facial expression, which can be either "mad", "weird" or "relaxed", as well as how many units of beer he has drunk that day. Each day he tries a hundred times to hit the bull's-eye, and we write down how often he succeeds. The data look like this:

  \begin{tabular}{rrrr} \multicolumn{1}{l}{} & & & \ Day & Beer & Expression & BullsEye \ 1 & 3 & Mad & 30 \ 2 & 3 & Mad & 32 \ $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \ 60 & 2 & Mad & 37 \ 61 & 4 & Weird & 30 \ $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \ 110 & 4 & Weird & 28 \ 111 & 2 & Relaxed & 35 \ $\vdots$ & $\vdots$ & $\vdots$ & $\vdots$ \ 150 & 3 & Relaxed & 31 \end{tabular}

  Write down a reasonable model for $Y_{1}, \ldots, Y_{n}$, where $n=150$ and where $Y_{i}$ is the number of times Barney has hit bull's-eye on the $i$ th day. Explain briefly why we may wish initially to include interactions between the variables. Write the $\mathrm{R}$ code to fit your model.

  The scientist of the above story fitted her own generalized linear model, and subsequently obtained the following summary (abbreviated):

  

  Why are ExpressionMad and Beer:ExpressionMad not listed? Suppose on a particular day, Barney's facial expression is weird, and he drank three units of beer. Give the linear predictor in the scientist's model for this day.

  Based on the summary, how could you improve your model? How could one fit this new model in $R$ (without modifying the data file)?

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• Paper 2, Section II, C

  #Statistical Physics

  2010

  Consider a 3-dimensional gas of $N$ non-interacting particles in a box of size $L$ where the allowed momenta are $\left\{\mathbf{p}_{i}\right\}$. Assuming the particles have an energy $\epsilon(|\mathbf{p}|), \epsilon^{\prime}(p)>0$, calculate the density of states $g(\epsilon) d \epsilon$ as $L \rightarrow \infty$.

  Treating the particles as classical explain why the partition function is

  $$Z=\frac{z^{N}}{N !}, \quad z=\int_{0}^{\infty} d \epsilon g(\epsilon) e^{-\epsilon / k T}$$

  Obtain an expression for the total energy $E$.

  Why is $\mathbf{p}_{i} \propto 1 / L ?$ By considering the dependence of the energies on the volume $V$ show that the pressure $P$ is given by

  $$P V=\frac{N}{3 z} \int_{0}^{\infty} d \epsilon g(\epsilon) p \epsilon^{\prime}(p) e^{-\epsilon / k T}$$

  What are the results for the pressure for non-relativistic particles and also for relativistic particles when their mass can be neglected?

  What is the thermal wavelength for non-relativistic particles? Why are the classical results correct if the thermal wavelength is much smaller than the mean particle separation?

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• Paper 3, Section II, C

  #Statistical Physics

  2010

  (i) Given the following density of states for a particle in 3 dimensions

  $$g(\varepsilon)=K V \varepsilon^{1 / 2}$$

  write down the partition function for a gas of $N$ such non-interacting particles, assuming they can be treated classically. From this expression, calculate the energy $E$ of the system and the heat capacities $C_{V}$ and $C_{P}$. You may take it as given that $P V=\frac{2}{3} E$.

  [Hint: The formula $\int_{0}^{\infty} d y y^{2} e^{-y^{2}}=\sqrt{\pi} / 4$ may be useful.]

  (ii) Using thermodynamic relations obtain the relation between heat capacities and compressibilities

  $$\frac{C_{P}}{C_{V}}=\frac{\kappa_{T}}{\kappa_{S}}$$

  where the isothermal and adiabatic compressibilities are given by

  $$\kappa=-\frac{1}{V} \frac{\partial V}{\partial P},$$

  derivatives taken at constant temperature and entropy, respectively.

  (iii) Find $\kappa_{T}$ and $\kappa_{S}$ for the ideal gas considered above.

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• Paper 4, Section II, C

  #Statistical Physics

  2010

  (i) Let $\rho_{i}$ be the probability that a system is in a state labelled by $i$ with $N_{i}$ particles and energy $E_{i}$. Define

  $$s\left(\rho_{i}\right)=-k \sum_{i} \rho_{i} \log \rho_{i} .$$

  $s\left(\rho_{i}\right)$ has a maximum, consistent with a fixed mean total number of particles $N$, mean total energy $E$ and $\sum_{i} \rho_{i}=1$, when $\rho_{i}=\bar{\rho}_{i}$. Let $S(E, N)=s\left(\bar{\rho}_{i}\right)$ and show that

  $$\frac{\partial S}{\partial E}=\frac{1}{T}, \quad \frac{\partial S}{\partial N}=-\frac{\mu}{T},$$

  where $T$ may be identified with the temperature and $\mu$ with the chemical potential.

  (ii) For two weakly coupled systems 1,2 then $\rho_{i, j}=\rho_{1, i} \rho_{2, j}$ and $E_{i, j}=E_{1, i}+E_{2, j}$, $N_{i, j}=N_{1, i}+N_{2, j}$. Show that $S(E, N)=S_{1}\left(E_{1}, N_{1}\right)+S_{2}\left(E_{2}, N_{2}\right)$ where, if $S(E, N)$ is stationary under variations in $E_{1}, E_{2}$ and $N_{1}, N_{2}$ for $E=E_{1}+E_{2}, N=N_{1}+N_{2}$ fixed, we must have $T_{1}=T_{2}, \mu_{1}=\mu_{2}$.

  (iii) Define the grand partition function $\mathcal{Z}(T, \mu)$ for the system in (i) and show that

  $$k \log \mathcal{Z}=S-\frac{1}{T} E+\frac{\mu}{T} N, \quad S=\frac{\partial}{\partial T}(k T \log \mathcal{Z})$$

  (iv) For a system with single particle energy levels $\epsilon_{r}$ the possible states are labelled by $i=\left\{n_{r}: n_{r}=0,1\right\}$, where $N_{i}=\sum_{r} n_{r}, E_{i}=\sum_{r} n_{r} \epsilon_{r}$ and $\sum_{i}=\prod_{r} \sum_{n_{r}=0,1}$. Show that

  $$\bar{\rho}_{i}=\prod_{r} \frac{e^{-n_{r}\left(\epsilon_{r}-\mu\right) / k T}}{1+e^{-\left(\epsilon_{r}-\mu\right) / k T}} .$$

  Calculate $\bar{n}_{r}$. How is this related to a free fermion gas?

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• Paper 1, Section II, I

  #Stochastic Financial Models

  2010

  What is a Brownian motion? State the reflection principle for Brownian motion.

  Let $W=\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion. Let $M=\max _{0 \leqslant t \leqslant 1} W_{t}$. Prove

  $$\mathbb{P}\left(M \geqslant x, W_{1} \leqslant x-y\right)=\mathbb{P}\left(M \geqslant x, W_{1} \geqslant x+y\right)$$

  for all $x, y \geqslant 0$. Hence, show that the random variables $M$ and $\left|W_{1}\right|$ have the same distribution.

  Find the density function of the random variable $R=W_{1} / M$.

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• Paper 2, Section II, I

  #Stochastic Financial Models

  2010

  What is a martingale? What is a supermartingale? What is a stopping time?

  Let $M=\left(M_{n}\right)_{n \geqslant 0}$ be a martingale and $\hat{M}=\left(\hat{M}_{n}\right)_{n \geqslant 0}$ a supermartingale with respect to a common filtration. If $M_{0}=\hat{M}_{0}$, show that $\mathbb{E} M_{T} \geqslant \mathbb{E} \hat{M}_{T}$ for any bounded stopping time $T$.

  [If you use a general result about supermartingales, you must prove it.]

  Consider a market with one stock with prices $S=\left(S_{n}\right)_{n \geqslant 0}$ and constant interest rate $r$. Explain why an investor's wealth $X$ satisfies

  $$X_{n}=(1+r) X_{n-1}+\pi_{n}\left[S_{n}-(1+r) S_{n-1}\right]$$

  where $\pi_{n}$ is the number of shares of the stock held during the $n$th period.

  Given an initial wealth $X_{0}$, an investor seeks to maximize $\mathbb{E} U\left(X_{N}\right)$ where $U$ is a given utility function. Suppose the stock price is such that $S_{n}=S_{n-1} \xi_{n}$ where $\left(\xi_{n}\right)_{n} \geqslant 1$ is a sequence of independent and identically distributed random variables. Let $V$ be defined inductively by

  $$V(n, x, s)=\sup _{p \in \mathbb{R}} \mathbb{E} V\left[n+1,(1+r) x-p s\left(1+r-\xi_{1}\right), s \xi_{1}\right]$$

  with terminal condition $V(N, x, s)=U(x)$ for all $x, s \in \mathbb{R}$.

  Show that the process $\left(V\left(n, X_{n}, S_{n}\right)\right)_{0 \leqslant n \leqslant N}$ is a supermartingale for any trading strategy $\pi$.

  Suppose $\pi^{*}$ is a trading strategy such that the corresponding wealth process $X^{*}$ makes $\left(V\left(n, X_{n}^{*}, S_{n}\right)\right)_{0} \leqslant n \leqslant N$ a martingale. Show that $\pi^{*}$ is optimal.

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• Paper 3, Section II, I

  #Stochastic Financial Models

  2010

  Consider a market with two assets, a riskless bond and a risky stock, both of whose initial (time-0) prices are $B_{0}=1=S_{0}$. At time 1 , the price of the bond is a constant $B_{1}=R>0$ and the price of the stock $S_{1}$ is uniformly distributed on the interval $[0, C]$ where $C>R$ is a constant.

  Describe the set of state price densities.

  Consider a contingent claim whose payout at time 1 is given by $S_{1}^{2}$. Use the fundamental theorem of asset pricing to show that, if there is no arbitrage, the initial price of the claim is larger than $R$ and smaller than $C$.

  Now consider an investor with initial wealth $X_{0}=1$, and assume $C=3 R$. The investor's goal is to maximize his expected utility of time-1 wealth $\mathbb{E} U\left[R+\pi\left(S_{1}-R\right)\right]$, where $U(x)=\sqrt{x}$. Show that the optimal number of shares of stock to hold is $\pi^{*}=1$.

  What would be the investor's marginal utility price of the contingent claim described above?

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• Paper 4, Section II, I

  #Stochastic Financial Models

  2010

  Consider a market with no riskless asset and $d$ risky stocks where the price of stock $i \in\{1, \ldots, d\}$ at time $t \in\{0,1\}$ is denoted $S_{t}^{i}$. We assume the vector $S_{0} \in \mathbb{R}^{d}$ is not random, and we let $\mu=\mathbb{E} S_{1}$ and $V=\mathbb{E}\left[\left(S_{1}-\mu\right)\left(S_{1}-\mu\right)^{T}\right]$. Assume $V$ is not singular.

  Suppose an investor has initial wealth $X_{0}=x$, which he invests in the $d$ stocks so that his wealth at time 1 is $X_{1}=\pi^{T} S_{1}$ for some $\pi \in \mathbb{R}^{d}$. He seeks to minimize the $\operatorname{var}\left(X_{1}\right)$ subject to his budget constraint and the condition that $\mathbb{E} X_{1}=m$ for a given constant $m \in \mathbb{R}$.

  Illustrate this investor's problem by drawing a diagram of the mean-variance efficient frontier. Write down the Lagrangian for the problem. Show that there are two vectors $\pi_{A}$ and $\pi_{B}$ (which do not depend on the constants $x$ and $m$ ) such that the investor's optimal portfolio is a linear combination of $\pi_{A}$ and $\pi_{B}$.

  Another investor with initial wealth $Y_{0}=y$ seeks to maximize $\mathbb{E} U\left(Y_{1}\right)$ his expected utility of time 1 wealth, subject to his budget constraint. Assuming that $S_{1}$ is Gaussian and $U(w)=-e^{-\gamma w}$ for a constant $\gamma>0$, show that the optimal portfolio in this case is also a linear combination of $\pi_{A}$ and $\pi_{B}$.

  [You may use the moment generating function of the Gaussian distribution without derivation.]

  Continue to assume $S_{1}$ is Gaussian, but now assume that $U$ is increasing, concave, and twice differentiable, and that $U, U^{\prime}$ and $U^{\prime \prime}$ are of exponential growth but not necessarily of the form $U(w)=-e^{-\gamma w}$. (Recall that a function $f$ is of exponential growth if $|f(w)| \leqslant a e^{b|w|}$ for some constants positive constants $a, b$.) Prove that the utility maximizing investor still holds a linear combination of $\pi_{A}$ and $\pi_{B}$.

  [You may use the Gaussian integration by parts formula

  $$\mathbb{E}[\nabla f(Z)]=\mathbb{E}[Z f(Z)]$$

  where $Z=\left(Z_{1}, \ldots, Z_{d}\right)^{T}$ is a vector of independent standard normal random variables, and $f$ is differentiable and of exponential growth. You may also interchange integration and differentiation without justification.]

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• Paper 1, Section I, F

  #Topics in Analysis

  2010

  Let $(X, d)$ be a non-empty complete metric space with no isolated points, $G$ an open dense subset of $X$ and $E$ a countable dense subset of $X$.

  (i) Stating clearly any standard theorem you use, prove that $G \backslash E$ is a dense subset of $X$.

  (ii) If $G$ is only assumed to be uncountable and dense in $X$, does it still follow that $G \backslash E$ is dense in $X$ ? Justify your answer.

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• Paper 2, Section I, F

  #Topics in Analysis

  2010

  (a) State the Weierstrass approximation theorem concerning continuous real functions on the closed interval $[0,1]$.

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous.

  (i) If $\int_{0}^{1} f(x) x^{n} d x=0$ for each $n=0,1,2, \ldots$, prove that $f$ is the zero function.

  (ii) If we only assume that $\int_{0}^{1} f(x) x^{2 n} d x=0$ for each $n=0,1,2, \ldots$, prove that it still follows that $f$ is the zero function.

  [If you use the Stone-Weierstrass theorem, you must prove it.]

  (iii) If we only assume that $\int_{0}^{1} f(x) x^{2 n+1} d x=0$ for each $n=0,1,2, \ldots$, does it still follow that $f$ is the zero function? Justify your answer.

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• Paper 3, Section I, F

  #Topics in Analysis

  2010

  Let $A=\{z \in \mathbb{C}: 1 / 2 \leqslant|z| \leqslant 2\}$ and suppose that $f$ is complex analytic on an open subset containing $A$.

  (i) Give an example, with justification, to show that there need not exist a sequence of complex polynomials converging to $f$ uniformly on $A$.

  (ii) Let $R \subset \mathbb{C}$ be the positive real axis and $B=A \backslash R$. Prove that there exists a sequence of complex polynomials $p_{1}, p_{2}, p_{3}, \ldots$ such that $p_{j} \rightarrow f$ uniformly on each compact subset of $B$.

  (iii) Let $p_{1}, p_{2}, p_{3}, \ldots$ be the sequence of polynomials in (ii). If this sequence converges uniformly on $A$, show that $\int_{C} f(z) d z=0$, where $C=\{z \in \mathbb{C}:|z|=1\}$.

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• Paper 4, Section I, $2 F$

  #Topics in Analysis

  2010

  Find explicitly a polynomial $p$ of degree $\leqslant 3$ such that

  $$\sup _{x \in[-1,1]}\left|x^{4}-p(x)\right| \leqslant \sup _{x \in[-1,1]}\left|x^{4}-q(x)\right|$$

  for every polynomial $q$ of degree $\leqslant 3$. Justify your answer.

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• Paper 2, Section II, 11F

  #Topics in Analysis

  2010

  Let

  $$B_{r}(0)=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}<r^{2}\right\},$$

  $B=B_{1}(0)$, and

  $$C=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$$

  Let $D=B \cup C$.

  (i) State the Brouwer fixed point theorem on the plane.

  (ii) Show that the Brouwer fixed point theorem on the plane is equivalent to the nonexistence of a continuous map $F: D \rightarrow C$ such that $F(p)=p$ for each $p \in C$.

  (iii) Let $G: D \rightarrow \mathbb{R}^{2}$ be continuous, $0<\epsilon<1$ and suppose that

  $$|p-G(p)|<\epsilon$$

  for each $p \in C$. Using the Brouwer fixed point theorem or otherwise, prove that

  $$B_{1-\epsilon}(0) \subseteq G(B)$$

  [Hint: argue by contradiction.]

  (iv) Let $q \in B$. Does there exist a continuous map $H: D \rightarrow \mathbb{R}^{2} \backslash\{q\}$ such that $H(p)=p$ for each $p \in C$ ? Justify your answer.

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• Paper 3, Section II, F

  #Topics in Analysis

  2010

  (i) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map with $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin.

  (ii) For $j=0,1$, let $\gamma_{j}:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma_{j}(0)=\gamma_{j}(1)$. Make the following statement precise, and prove it: if $\gamma_{0}$ can be continuously deformed into $\gamma_{1}$ through a family of continuous curves missing the origin, then $w\left(\gamma_{0} ; 0\right)=w\left(\gamma_{1} ; 0\right)$.

  [You may use without proof the following fact: if $\gamma, \delta:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ are continuous with $\gamma(0)=\gamma(1), \delta(0)=\delta(1)$ and if $|\gamma(t)|<|\delta(t)|$ for each $t \in[0,1]$, then $w(\gamma+\delta ; 0)=w(\delta ; 0)$.]

  (iii) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma(0)=\gamma(1)$. If $\gamma(t)$ is not equal to a negative real number for each $t \in[0,1]$, prove that $w(\gamma ; 0)=0$.

  (iv) Let $D=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and $C=\{z \in \mathbb{C}:|z|=1\}$. If $g: D \rightarrow C$ is continuous, prove that for each non-zero integer $n$, there is at least one point $z \in C$ such that $z^{n}+g(z)=0$.

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• Paper 1, Section II, A

  #Waves

  2010

  Derive the wave equation governing the velocity potential $\phi$ for linearized sound waves in a compressible inviscid fluid. How is the pressure disturbance related to the velocity potential?

  A semi-infinite straight tube of uniform cross-section is aligned along the positive $x$-axis with its end at $x=-L$. The tube is filled with fluid of density $\rho_{1}$ and sound speed $c_{1}$ in $-L<x<0$ and with fluid of density $\rho_{2}$ and sound speed $c_{2}$ in $x>0$. A piston at the end of the tube performs small oscillations such that its position is $x=-L+\epsilon e^{i \omega t}$, with $\epsilon \ll L$ and $\epsilon \omega \ll c_{1}, c_{2}$. Show that the complex amplitude of the velocity potential in $x>0$ is

  $$-\epsilon c_{1}\left(\frac{c_{1}}{c_{2}} \cos \frac{\omega L}{c_{1}}+i \frac{\rho_{2}}{\rho_{1}} \sin \frac{\omega L}{c_{1}}\right)^{-1} .$$

  Calculate the time-averaged acoustic energy flux in $x>0$. Comment briefly on the variation of this result with $L$ for the particular case $\rho_{2} \ll \rho_{1}$ and $c_{2}=\mathrm{O}\left(c_{1}\right)$.

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• Paper 2, Section II, 38A

  #Waves

  2010

  The equation of motion for small displacements $\mathbf{u}(\mathbf{x}, t)$ in a homogeneous, isotropic, elastic medium of density $\rho$ is

  $$\rho \frac{\partial^{2} \mathbf{u}}{\partial t^{2}}=(\lambda+\mu) \boldsymbol{\nabla}(\boldsymbol{\nabla} \cdot \mathbf{u})+\mu \nabla^{2} \mathbf{u}$$

  where $\lambda$ and $\mu$ are the Lamé constants. Show that the dilatation $\nabla \cdot \mathbf{u}$ and rotation $\nabla \wedge \mathbf{u}$ each satisfy wave equations, and determine the corresponding wave speeds $c_{P}$ and $c_{S}$.

  Show also that a solution of the form $\mathbf{u}=\mathbf{A} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$ satisfies

  $$\omega^{2} \mathbf{A}=c_{P}^{2} \mathbf{k}(\mathbf{k} \cdot \mathbf{A})-c_{S}^{2} \mathbf{k} \wedge(\mathbf{k} \wedge \mathbf{A})$$

  Deduce the dispersion relation and the direction of polarization relative to $\mathbf{k}$ for plane harmonic $P$-waves and plane harmonic $S$-waves.

  Now suppose the medium occupies the half-space $z \leqslant 0$ and that the boundary $z=0$ is stress free. Show that it is possible to find a self-sustained combination of evanescent $P$-waves and $S V$-waves (i.e. a Rayleigh wave), proportional to exp $[i k(x-c t)]$ and propagating along the boundary, provided the wavespeed $c$ satisfies

  $$\left(2-\frac{c^{2}}{c_{S}^{2}}\right)^{2}=4\left(1-\frac{c^{2}}{c_{S}^{2}}\right)^{1 / 2}\left(1-\frac{c^{2}}{c_{P}^{2}}\right)^{1 / 2}$$

  [You are not required to show that this equation has a solution.]

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• Paper 3, Section II, 38A

  #Waves

  2010

  Consider the equation

  $$\frac{\partial^{2} \phi}{\partial t \partial x}=-\alpha \phi$$

  where $\alpha$ is a positive constant. Find the dispersion relation for waves of frequency $\omega$ and wavenumber $k$. Sketch graphs of the phase velocity $c(k)$ and the group velocity $c_{g}(k)$.

  A disturbance localized near $x=0$ at $t=0$ evolves into a dispersing wave packet. Will the wavelength and frequency of the waves passing a stationary observer located at a large positive value of $x$ increase or decrease for $t>0$ ? In which direction do the crests pass the observer?

  Write down the solution $\phi(x, t)$ with initial value

  $$\phi(x, 0)=\int_{-\infty}^{\infty} A(k) e^{i k x} d k$$

  What can be said about $A(-k)$ if $\phi$ is real?

  Use the method of stationary phase to obtain an approximation for $\phi(V t, t)$ for fixed $V>0$ and large $t$. What can be said about the solution at $x=-V t$ for large $t$ ?

  [You may assume that $\int_{-\infty}^{\infty} e^{-a u^{2}} d u=\sqrt{\frac{\pi}{a}}$ for $\operatorname{Re}(a) \geqslant 0, a \neq 0$.]

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• Paper 4, Section II, A

  #Waves

  2010

  Starting from the equations for one-dimensional unsteady flow of an inviscid compressible fluid, show that it is possible to find Riemann invariants $u \pm Q$ that are constant on characteristics $C_{\pm}$given by

  $$\frac{d x}{d t}=u \pm c$$

  where $u(x, t)$ is the velocity of the fluid and $c(x, t)$ is the local speed of sound. Show that $Q=2\left(c-c_{0}\right) /(\gamma-1)$ for the case of a perfect gas with adiabatic equation of state $p=p_{0}\left(\rho / \rho_{0}\right)^{\gamma}$, where $p_{0}, \rho_{0}$ and $\gamma$ are constants, $\gamma>1$ and $c=c_{0}$ when $\rho=\rho_{0}$.

  Such a gas initially occupies the region $x>0$ to the right of a piston in an infinitely long tube. The gas is initially uniform and at rest with density $\rho_{0}$. At $t=0$ the piston starts moving to the left at a constant speed $V$. Assuming that the gas keeps up with the piston, find $u(x, t)$ and $c(x, t)$ in each of the three distinct regions that are defined by families of $C_{+}$characteristics.

  Now assume that the gas does not keep up with the piston. Show that the gas particle at $x=x_{0}$ when $t=0$ follows a trajectory given, for $t>x_{0} / c_{0}$, by

  $$x(t)=\frac{\gamma+1}{\gamma-1}\left(\frac{c_{0} t}{x_{0}}\right)^{2 /(\gamma+1)} x_{0}-\frac{2 c_{0} t}{\gamma-1}$$

  Deduce that the velocity of any given particle tends to $-2 c_{0} /(\gamma-1)$ as $t \rightarrow \infty$.

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• Paper 3, Section I, D

  #Variational Principles

  2010

  Derive the Euler-Lagrange equation for the function $u(x, y)$ which gives a stationary value of

  $$I=\int_{\mathcal{D}} L\left(x, y, u, \frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}\right) d x d y$$

  where $\mathcal{D}$ is a bounded domain in the $(x, y)$ plane, with $u$ fixed on the boundary $\partial \mathcal{D}$.

  Find the equation satisfied by the function $u$ which gives a stationary value of

  $$I=\int_{\mathcal{D}}\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial u}{\partial y}\right)^{2}\right] d x d y$$

  with $u$ given on $\partial \mathcal{D}$.

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• Paper 2, Section II, D

  #Variational Principles

  2010

  Describe briefly the method of Lagrange multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $\phi(x, y)=0$.

  A tent manufacturer wants to maximize the volume of a new design of tent, subject only to a constant weight (which is directly proportional to the amount of fabric used). The models considered have either equilateral-triangular or semi-circular vertical crosssection, with vertical planar ends in both cases and with floors of the same fabric. Which shape maximizes the volume for a given area $A$ of fabric?

  [Hint: $(2 \pi)^{-1 / 2} 3^{-3 / 4}(2+\pi)<1 .$ ]

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• Paper 4, Section II, D

  #Variational Principles

  2010

  A function $\theta(\phi)$ with given values of $\theta\left(\phi_{1}\right)$ and $\theta\left(\phi_{2}\right)$ makes the integral

  $$I=\int_{\phi_{1}}^{\phi_{2}} \mathcal{L}\left(\theta, \theta^{\prime}\right) d \phi$$

  stationary with respect to small variations of $\theta$ which vanish at $\phi_{1}$ and $\phi_{2}$. Show that $\theta(\phi)$ satisfies the first integral of the Euler-Lagrange equation,

  $$\mathcal{L}\left(\theta, \theta^{\prime}\right)-\theta^{\prime}\left(\partial \mathcal{L} / \partial \theta^{\prime}\right)=C,$$

  for some constant $C$. You may state the Euler-Lagrange equation without proof.

  It is desired to tow an iceberg across open ocean from a point on the Antarctic coast (longitude $\phi_{1}$ ) to a place in Australia (longitude $\phi_{2}$ ), to provide fresh water for irrigation. The iceberg will melt at a rate proportional to the difference between its temperature (the constant $T_{0}$, measured in degrees Celsius and therefore negative) and the sea temperature $T(\theta)>T_{0}$, where $\theta$ is the colatitude (the usual spherical polar coordinate $\left.\theta\right)$. Assume that the iceberg is towed at a constant speed along a path $\theta(\phi)$, where $\phi$ is the longitude. Given that the infinitesimal arc length on the unit sphere is $\left(d \theta^{2}+\sin ^{2} \theta d \phi^{2}\right)^{1 / 2}$, show that the total ice melted along the path from $\phi_{1}$ to $\phi_{2}$ is proportional to

  $$I=\int_{\phi_{1}}^{\phi_{2}}\left(T(\theta)-T_{0}\right)\left(\theta^{\prime 2}+\sin ^{2} \theta\right)^{1 / 2} d \phi$$

  Now suppose that in the relevant latitudes, the sea temperature may be approximated by $T(\theta)=T_{0}(1+3 \tan \theta)$. (Note that $(1+3 \tan \theta)$ is negative in the relevant latitudes.) Show that any smooth path $\theta(\phi)$ which minimizes the total ice melted must satisfy

  $$\theta^{\prime 2}=\sin ^{2} \theta\left(\frac{1}{4} k^{2} \tan ^{2} \theta \sin ^{2} \theta-1\right),$$

  and hence that

  $$\sin ^{2} \theta=\frac{2}{1-\left(1+k^{2}\right)^{1 / 2} \sin 2\left(\phi-\phi_{0}\right)},$$

  where $k$ and $\phi_{0}$ are constants.

  [Hint:

  $$\int \frac{d x}{x\left(\alpha^{2} x^{4}+x^{2}-1\right)^{1 / 2}}=\frac{1}{2} \arcsin \left[\frac{x^{2}-2}{x^{2}\left(1+4 \alpha^{2}\right)^{1 / 2}}\right]$$

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